Jolley to align pages correctly

1 SUMMATION OF SERIES COLLECTED BY L.B.W. JOLLEY, M.A. (CANTAB.), M.LE.E. Seconp Revisep Epttion DOVER PUBLICATIONS, INC. NEW YORK j Copyright ©1961 byDover Publications, Inc Alrights reserved under Pan American and In- ternational Copyright Conventions. Published inCanada byGeneral Publishing Com- pany, Ltd,, 30Lesmill Road, Don Mills, Toronto, Ontario. Published inthe United Kingdom byConstable and Company, Ltd., 10Orange Street, London we 2. This Dover edition, first published in1961, isa revised and enlarged version ofthework first pub- lished byChapman &Hall, Ltd., in1925, Library ofCongrest Catalog Card Number: 61-68274 International Standard Book Number: 0-486-60023-8 ‘Manufactured inthe United States ofAmerica Dover Publications, Ine. 180 Varick Street New York, N. ¥.10014 ! PREFACE TO DOVER EDITION Aseconp edition published inthe United States ofAmerica provides anopportunity forincluding many new series with an increase ofmore than 50percent over theoriginal number. It hasalso been possible torearrange theseries inamore reasonable form. ‘Some corrections have been received from readers, and useful suggestions have been made bythem forthepresent arrangement. These aregratefully acknowledged, and itwillbeofgreat assist- ance for future editions ifreaders will communicate their ideas forfurther expansion. Inusing this collection himself, the author has experienced difficulty intracing certain series, and there does not seem any solution excepting acomplete search through alltheseries given. For example, certain series including inverse products appear in different parts ofthe book, and, ifasearch istobeavoided, a complete rearrangement combined with excessive duplication would benecessary. Itdoes notseem possible, asinthecase of acollection ofintegrals, toarrange them inacompletely rational manner. Any suggestion inthis direction would bespecially welcome. ‘Among the new series included aresome ofthose developed byGlaisher inmany publications, notably theQuarterly Journal ofMathematics. Basing bis series onBernoulli functions, Glaisher evolved anumber ofcoefficients which apparently simplify theappearance oftheseries. Inthis present collection, only afewaregiven, andtheoriginal articles should beconsulted ifthereaderwishestoinvestigate themfurther. The author wishes toacknowledge permission bytheLondon Scientific Computing Service topublish tables from theIndex of Mathematical Tables byFletcher, Miller, and Rosenhead—an vi PREFACE TO DOVER EDITION exceedingly useful book for any engaged inwork onapplied mathematics; and also tothank Mrs. H.M.Cooper for her excellent work intyping adifficult manuscript. L.B.W. Jouey 623 Upper Richmond Road West, Richmond, Surrey, 1960 | PREFACE TO FIRST EDITION For along time past there hasbeen aneed foracollection of series into one small volume foreasy reference together with a bibliography indicating atleast one ofthetextbooks towhich reference could bemade incase ofdoubt astoaccuracy ortothe method bywhich theseries was arrived at. The 700-odd series inthis collection (with theexception ofa fewwhich have been specially prepared) arenotnew, and repre sent only thelabour ofextracting thematerial from the many textbooks onalgebra, trigonometry, calculus and thelike. Yetsuchacollection will,itisfelt,beofconsiderable benefittothoseengaged inthesolution oftechnical problems, and will save a greatdealoftimeinsearching fortherequired result. Criticism may beoffered onthegrounds that theinclusion of easy algebraical summations isunnecessary, butthey have been inserted foravery definite purpose. For example, aseries of inverse products may have foritssum anexpression which is simple tofind; butontheother hand, thesolution may entail a complicated expression involving theintegration ordifferentiation ofotherseries. Forthisreasonthearrangement oftheserieshasbeen difficult, andoverlapping isunavoidable incertain instances. Toovercome thisdifficulty, theseries have been setforth inas pictorial amanner aspossible, sothattheform oftheindividual terms can bereadily seen. Onthis account also, theinclusion ofsuch series asareevolved forelliptic integrals, Bessel functions and the like has been restricted, perhaps totoogreat anextent; butreference tostandard works isusually essential insuch cases, andpractically only such references are included. The final column refers tothebibliography atthebeginning of thebook, and here again ithasbeen quite impossible forobvious reasons toprovide forallthereferences. viii PREFACE TOFIRSTEDITION One ofthemost useful works, ifitisdesired topursue any one particular problem further, istheSmithsonian Tables. The scope ofmany oftheseries can begreatly enlarged by differentiation orintegration ofsome oftheforms given, and in thecase ofanintegrated series, theconstant ofintegration must, ‘beobtained bysuitable methods. Infinite products areoften of value inobtaining new series bytaking logarithms and by differentiating orintegrating subsequently. Inmany cases ithasbeen impossible inthissmall volume to comment onthelimits orassumptions made inany particular summation; particularly isthis thecase with oscillating series: and incase ofdoubt itisalways safer torefer toa textbook, and tobear inmind that thiscollection issupplementary to,and not inplace of,theusual mathematical books. Special attention isdrawn incases ofdifficult summations to theGeneral and Special Forms (pages 216-225). Inallcases logh denotes thelogarithm totheNapierian base, inaccordance with modern practice. Finally, any additions orcorrections would bewelcomed for embodiment insubsequent editions. L.B.W. Joutey Fairdene, Sheen Road, Richmond, Surrey, 1925 CONTENTS Series No. Page 1.ARITHMETICAL PROGRESSION . . 1 2 TI. GEOMETRICAL PROGRESSION . . 2 2 TIT. ARITHMETICAL AND GEOMETRICAL PROGRESSION. ....52 TV. PoWERS OFNATURAL NUMBERS . 17 4 VY. Propucts OFNATURAL NUMBERS =. 42 8 VI.FIGURATE ANDPOLYGONAL NUMBERS. 60 12 VII. INVERSE NATURAL NUMBERS. . 70 14 VIII. EXPONENTIAL AND LOGARITHMIC SERIES 7 18 TX. BINOMIALS . . . . 165 32 ‘X. Simpce INveRsE PRODUCTS . . 201 38 XI. OTHER INVERSE PRODUCTS . . 233 44 XI. SIMPLE Factoriats. =. 0. 82 SD. XII. OrHeR Power Series (Bernoulli's and Euler's numbers)... .29252 XIV. TRIGONOMETRICAL SUMMATIONS . . 417 B XY. HYPERBOLIC SUMMATIONS . . 7 134 XVI. TRIGONOMETRICAL EXPANSIONS... 732,138 XVII. HYPERBOLIC EXPANSIONS . . 871 162 XVIII. TayLor’s AND MACLAURIN’S THEOREM 957 178 XIX. BESSEL FUNCTIONS . . . 959 178 XX. ELuiptic FUNCTIONS. . . 967 178 XXI. Various INTEGRALS . . . 969 180 XXII. Beta AND GAMMA FUNCTIONS . .1008 186 XXII. Inrinrre Propucts . . .1016 188 XXIV. Fourier’s Serres - . . .1085 200 XXYV. HYPERGEOMETRIC FUNCTIONS. .1090 202 XXXVI. RELATIONS BETWEEN PRODUCTS AND ‘SERIES .....1094204 XXVII. SPECIAL FUNCTIONS . . . .H0l 206 x CONTENTS SeriesNo.Page XXVIIL Zeta Functions... ‘1103212 XXIX. Lecenore Potynomiats . =. =.‘1104214 XXX. SPECIAL Propucts . . =» =.«0S. 214 XXXI. GENERAL FORMS 5. =1106 216 XXXII Douate AND Teste Serms . .118224 XXXII. BERNOULL’s FUNCTIONS . 1128 226 Bernoulli's Numbers . . .‘1129288 Table of Bernoulli's Numbers inVulgar Fractions... 230 ‘Table ofBernoulli's Numbers inIntegers and Repeating Decimals. - ee 232 Values ofConstants in Series (805)to(318)and(1130). 234 Euler's Numbers... LISI (288 Euler's Constant... .1182288 Sum ofPower Series . . .1183 240 Relations between Bernoulli’s Numbers... «1S 242 BIBLIOGRAPHY Indiatng Author TitleandPublisherAt—T.J.Bromwich Introduction totheTheoryofInfiniteSeries, London: Macmillan Co., 1926. Bo LLL. Smail Elements ofthe Theory ofInfinite Processes, New York: McGraw- Hill Book Co., 1923. CG. Chrystal Algebra, AnElementary Text Book ‘fortheHigher Classes ofSecond- ‘arySchools, New York: Dover Publications,Inc.,1961. D___LevettandDavi-PlaneTrigonometry, NewYork: sonMacmillan Co., 1892. ES. L.Loney Plane Trigonometry (Parts1and1), Cambridge: Cambridge Univer- sity Press, 1900. FH. S.Hall and Higher Algebra, London: Mac- S.R. Knight millan Co., 1899. GE. T.Whittaker Calculus ofObservations, Glasgow: andG. Robinson Blackie and Son, 1937. HH. Lamb Infinitesimal Calculus, Cambridge:Cambridge University Press,1921. J L.Todhunter Integral Calculus, London: Mac- millan Co., 1880. K CP Steinmetz Engineering Mathematics, New York: McGraw-Hill Book Co., 1911. LJ. Edwards Differential Calculus forBeginners, London: Macmillan Co., 1899. M J.Edwards Integral Calculus for Beginners, London: Macmillan Co., 1898. N G.S. Carr Synopsis ofPure Mathematics, London: Hodgson, 1886. fInthetext, thenumbers preceding theReference letter refer tothe volume ofthework cited: the numbers following theReference letter refer topages. xii BIBLIOGRAPHYAndiating Author TitleandPublisher ter © E,W. Hobson 4Treatise onPlane Trigonometry, New York: Dover Publications, Ine., 1957, P Encyclopaedia Britannica, 11thedi- tion, Q —_E.T. Whittaker Modern Analysis, Cambridge: Cam-andG.N.Watson bridgeUniversityPress,1920.RE. Goursat ACourse inMathematical Analysis, Vol. 1,New York: Dover Pub- lications, Inc., 1959. T —_E.P.Adams Smithsonian Mathematical Formu- fae,Washington: "Smithsonian Institute, 1922. U_——W.E. Byerly Fourier’s Series, New York: Dover Publications, Inc., 1959.WG. Boole Calculus ofFinite—Differences,‘NewYork: Dover,1960. X A.Eagle Fourier's. Theorem, New York: Longmans Green’ and Co., 1925.YJ.Edwards Differential Calculus, London:Macmillan Co., 1938, Z J.Edwards Integral Calculus, Vols. 1and W, London: Macmillan Co., 1922. AA K.Knopp Theory andApplications ofInfinite Series, Glasgow: Blackie and Son, 1928. AB H.S.Carslaw —Fourier’s Series and Integrals, New York: Dover Publications, Inc., 1950. ACFletcher, Miller,IndexofMathematical Tables,Lon-andRosenhead "don: Scientific Computing Ser- vice, 1946. ADE. Jahnke and F, Tables ofFunctions, New York: Emde DoverPublications, Inc.,1945, AE J.W.L.Glaisher Quarterly Journal ofMathematics, ‘Vol. 29, 1898. AE. J.W.L. Glaisher Quarterly Journal ofMathematics, ‘Vol. 28, 1896. AGE.W.Hobson TheTheoryofFunctions ofaRealVariable, Vol. Ul, New York: Dover Publications, Inc., 1957. SUMMATION OF SERIES 2 SUMMATION OF SERIES SeriesNo1.Arithmetical Progression (1)a+(@+d)+(@+2d)+...nterms I.Geometrical Progression (2)a+ar+ar?+...nterms Q)atart+ar+...0 (4)1+ax+ax? +a8 +... TIL. Arithmetical and Geometrical Progression (5)a+(a+dr+(a+Ady?+...nterms (a+ (a+dy+(atdy+...0 (7)1+2x+3x2+4x3+... 4.7 10 @l+gtytyt... terms Ot ZtZtfe. terms 35,7 (QO)1+5+gtgt-- oO (11)1+3x+6x?+10x3+... £See footnote toBibliography. 1 ARITHMETICAL, GEOMETRICAL PROGRESSION 3 Reference =32a(n—Id}=3(a+1)wheres=lasttermF.29 =o) F.39 = wherer <1 F.40 =rewhereax<1 F1S8 8 dlr)fatn=Dyn TTF Tr Fas =poetapwhee F.4d =a wherex<1 Fag 35 12n+7 =767 16Se1 F.45 1 n -4-s5-5 F.45 =6 F.45 1=GapWeerer<1 F.45 4 SUMMATION OF SERIES SeriesNo.(12)xx+y)+2°?+y2)+8G+y*)+...terms 2,3,2,3,2ODF+pt pt RtPte 45,4 55 (95-Rt BAH Re RB (19)2++5+ayte-terms (16)1+3x+Sx?4...Qn=Dart 1V.PowersofNaturalNumbers (11)>x (18) 14243444000 (9) P+ BER EME MP OH)PEDEPH MHD (21) 14 26+ 38+ AEE ot | POWERS OF NATURAL NUMBERS 3 Reference_202"=1),yey"=1)=—3IT +“pol F.46 9 =3 F.46 =3 one F.46B = nt 1 =6MGn?+ntIDG _Lx=Qn+Dxe+(Qn—Dxott ~=F a NPY net 1(P) Baneprit2 +3(t) a"‘3(5)2, +3(8)By—...where(?)arethebinomial coefficients and B,are Bernoulli numbers, see No. (1129). The series ends with theterm innifpiseven, and with theterm inn?if pis odd. T.2 _a(n+1)-mapy F.50 _nn+Qn+1) F.50x . =fol+)?-f} FS =aonn+DQn+Gn+3n=1) F.256 6 SUMMATION OF SERIES SeriesNo. (22) 1+ 4HASH WS (23)16+26+36+464... 06 (24) 1727+ THAT HLL (5)P+2+2+P+... nterms 26B+3+934P+... mterms QDP+(15)+23+25+.(EY (28)2?+4?+62+82+...mterms (29) 12-21 +22.22 +32.23 +... mterms (30) 1-22+2-32+3-42+...mterms QI) (n2—12)+2G?—22)+3G?-32)+...mterms oyS@r- 9San o)Sormoet=n) (5)de+yor—9) I POWERS OF NATURAL NUMBERS 7 Reference nonS|Stn?E4548 F.337 nt on WS on ate eee F338 m8on?|In8Intnt<S+o+R-Bth F.338 =Sdn?-) F.256 =(Qn? —1) F.256 LL{@+Dima? 1 a1 ance et _2nln +1)Qn +1) ~ 3 =2{2n? —4n+6}-6 =Fyne+hn+2100+5) F.256 =4-1) F.323 =$nQn+0) =m2(Q2n? —1) meat=GD innsQn+1) =2G=)_AY-I)xT yal eV. Ly_(ey=Gy+1}_{6) HE)" + corey=1 0-3 x 8 SUMMATION OF SERIES Series No. aySoantl—yet ren)2S 38)>aon+ay!+...a)z SeeNo. (17) G9)2" 40)Dae o G1)D+ 243...+ met 7 YV.ProduetsofNaturalNumbers(42)Tofindthesumofntermsofaseries,eachtermofwhichiscomposed ofrfactors inarithmetical progression, the first factors oftheseveral terms being inthesame arith- metical progression: Write down thenth term, affix thenextfactorattheend,dividebythenumberoffactors thus increased, and bythe common difference, and add a constant. (43) 1-3-5 43-5-745-7.9 +...mterms (44) 1-2 +23+3-4...nterms (45) 2-5 +5-8 +8-11 +...mterms (46) 2-2 +4-4+7-8+11-16+16-32... mterms (AD) 1-2-3 423-4 +3-4-5... nterms (48) 1-2-3-4 +2-3-4-5 +...mterms I PRODUCTS OF NATURAL NUMBERS 9 Reference ==)_y=) @= y= 9 &= T= ={agb,+ayb,-1+...+an} [ee rees =poywherex<1 T-x -a>wherex<1 Lee _mntDetnln+Itt -G=— 720a +3 F.314 =Qn=Nn+NOn+Qn+5)15rr > as =n(2n3 +8n?+Tn—2) F315 =meeDoe F532 =nGn?+6n+1) F.318=—n4+4-4 F333 =fr+Dl+2K+3) F322 =Ela+In+2X+3X+4) F322, 10 SUMMATION OF SERIES SeriesNo. (49) 1-4-7 +4-7-10+7-10-13+...mterms (60) 1-4-7+2-58+3-6.9+...mterms (SI) 1-5-9 +2-6-10+3-7-11+...mterms (52) 6-9 +12-21 +20-37 +30-57 +...mterms nthterm is(n+In+2)(2n? +6n+1) (53) 2-2+6-4+12-8+20-16+30-32+...mtermsnthterm isn(n +1)2* (54) 1-3-2 +2-432 +3.5.42 +...mterms (55)>(p-nq-0) 7 (56)Smt Dent nD) cy S bb+NO+2)...+n=1) onDae+a+2)...@+n—1) ex 2x? (98)Sy +Fg +Mterms This series isintegrable ifx=4, StanbdLares+Dore 3 i PRODUCTS OF NATURAL NUMBERS II Reference =Gn~2Gn+NGn+Hn+7)+7 F322 =ntfn+On+7) F322 =Fmd+Din+80+9) F322 =2a(n+Dn+ln+3)+4) $l+DetHn+3)2 F331 =(ne QI 4 F332 =hme+Din+Den+3)+3) F.323 =MADant1)3p+9)+09 =GeWertDmetya—) | C.200 m+n,tn . bb+1b+2)... +n) __ 6 (+1 aaa +Ifa +2)...(a@+n-1) b+1-a 1.28 S nextDar TyF2) w58 at n-1 2 -Sne2t ws 2 SUMMATION OF SERIES Series No. VI.Figurate andPolygonal Numbers (60) Figurate numbers— insOeee ee12 3 4 5 6... 13 6 0 15 2.. 14 0 20 35 56... 1 5 15 35° 7 126... The sum tonterms ofthe rth order (61)MethodofDifferences due&hewntonlupomme One Seriesis 12 40 90 168 280432... IstDiff 2%5078112182...2ndDitt. 2B&BM4...3rd Diff. 6 6 6... 4th Diff. a) The mth term is The sum 12+2 4—Ye=2 ln 2yn—3) *3 (62)4+14+30-452+80+114+...mterms (63)8+26+54+92+140+198+...nterms (64)9+16+29-454-4...mterms (65)4+13435+94+262+...mterms (66) 2+ 12+36+80-4 150+252+...mterms 1 FIGURATE AND POLYGONAL NUMBERS. 13 Reference =Hen+Wn+2)...dn+r=1) F320 angémévo\ pour toutes les 1A silovdve delwpa ert par enonple des dlovs lemidme tevme estumequation du Keijad ye,domeduSeand degre =ian+BAD 4,Be—D4=2)FE 44x(n=In~2m~3) Yba ae _—_— Saw vevitier =BM +Gn? +230+46) F.326 =nn+iy =>pas bonne! F.332 =Fan+160+7) F.332 =62"=1)+bain+5) F333 3 1=FO-D+fmtNesy—n F333 =yma+Din+2X30+1) F.332 4 SUMMATION OF SERIES SeresNo, (67)30+144+420+960+1890+...mterms (68) 245413 +35+...terms (9)2+Tx+25x2+91x39+...mterms VIL.Inverse Natural Numbers (0)S04 toghnsd- an Bi aD -——43_- mnt Im 2) Also+54...44a Totiigt M1-Z44-f4$-.@ 1o1,iit M1-Z-F4h4}-..0 IN ftv ala a a(1-3-3) +6-5-a)+6-w-a) +2 5 14 9oat gt © lj11it 9)1-34 5-445-- ) 1 4 1(33+ seat ot) INVERSE NATURAL NUMBERS 1s Reference =ann+In+2m+34+21) F,332 =3G-D4r-1 F272 L401 -Jr Toa +Tae Fam where C=Euler's constant, seeNo.(1132), en re reBTB= w=TH=3 1aye, T.2 anda,=Zftx(t=92-2)... -1=) 27 =7.48547 wheren=1000 A.325 =1439273 where n=106 =logh2 =og2=LN& F.195 =0.43882 =7—Hlogh2 #475 =jlogh2 C252 < on—4=2Gree=m~8h C.283 =logh4 c.252 Loyarithme hyperboliqe ( —logarithme matyrel SGmQ+ fsEx1,U8L9 logarihme meperien 16 SUMMATION OF SERIES SeriesNo.(91435 -3-Ftgt aw -1-2fb+taipt 9)1-444-ht..0 0}-E45-B 4.0 @D1-F44-$+7-g4. 2)1-545-Pt...@ )1-f-Ft bt popte® 1-f44- ped-bee coreeeones G1-$412 123, Say , INVERSE NATURAL NUMBERS 7 Reference =r 7Mo710735 ©.335 -j OCNBSAIMNEGAS E109 =F+v2 O,AUBOS44YQ M132 =}(&+logh2) 33964 BBHBS A.189 =;(S=logh2)O,2ZISSSOTLTT —ais9 “iA ©,Lo4s9972K77P-281 =anfr+2logh(V2+1]O,BECAZIS7TY A.190 -Jaloen2+v3)O1E034519%61 AB.166 ns ©=a? 0,306844 682 A.528 =logh(1+V2) 100” Y.90 2 1+v3) =Jyloen(1234) Y.90 =}(C+loghn)+logh2+Bt-CTF 4. For Csee No. (1132) and forB,seeNo.(1129) A.325 18 SUMMATION OF SERIES SeriesNo. ax 4xt )iytigettree 2? 9)atatatnterms xt 0)atatte 1 2 4 OOsitweitee? 1-2& 2x—4x3 4x3—8x7 ©(ee ticegatioewe at® 11 11 1a ©«f-teaitase- eet] le te 1 |(4)$$.-154+$8—... 12x 12 1:3 2x_\s 05)(Fa)+ralrFa)+2ea(rs a)+© ¥ly 1 092a -aap} : 2 =-_@ -_-A=-5 Bwep Ccx ‘VIII.Exponential andLogarithmic Series 3 ON1+artSE4eP4 (98)1+xlogha+EIEN5op eox 09)x-F 4... =40-2) +HU -x)+... H EXPONENTIAL AND LOGARITHMIC SERIES 19 Reference x =[oy whee < Tus 1 1 “joe T=* A.24 =X wherex?<1,and l-x" --4 wherex>1 Ts x-1 =k wherex2>1 T.18 x-T 14x-Teepe Whee<i Y.54 wtp 4 nt. A.102 “x;+5@aD ties DEFT i. ~reg?where$2145454.0042 A.192 =x where|x|<1 Q.132 =dee AB.167 =e F.188 =a F.188 =logh(1+x)wherex<1 F.191 ! 20 SUMMATION OFSERIES SeriesNo,(100)=x-F-E-...0 Gon1424 F4FaFew (103)ptataee. o} (09>eat (05)@1)-F@-1F +5-H... 0097+54+84aBae 007«(I+x4FretasBat...co) (08)142+FEE FEISw 09)14Py eyee any2{F43(143)+5(t45+4)eeof 1 EXPONENTIAL AND LOGARITHMIC SERIES 21 Reference =logh (1—x) wherex<1 F191 =ISe F.339 =logh 3—logh 2 F.195 ea F.196 3 =32logh 2 .253 =loghx where0<x<2 TmLopdee[htds(Gott14,4, eens =xe-1) =7AG11a) =ee338 ' ?T L =e we 7.126 — | =( +e F.338 =et—logh (I+x) F,338 =flogh(1+2}?wherex<1 F191 =flogh(1—x)P A.191 ==logh(1+x)-logh(1—x) A191 =}(ant)logh+ A191 ! 2 SUMMATION OFSERIES SertesNo.SF(4) ot Oo Cw (115) 3a,e (6)1-3(1+5)+3(1+34{)-..0 amSh (118)Se c ay35 oyDay cayx+(1+p)ee(i tptger.. © (122)SS 023)xe+2a+.) (125)Somgta ! EXPONENTIAL AND LOGARITHMIC SERIES 23 Reference _Mu —prayBiiregare TQ) =3.6256 7.146 =4whereA=oand agi =(4+aeteBe MO You mil =F- plosh2 ©,SE2MMNOSLET A,520 x =logh5 H.460 logh wh 1 =loghpLwherex< ae-1 Ayiaye \ 1 1 =1=(5-1)toshts F.338 =pylon wheres <1Toa _(Gt =ax4Jer —bt—3~tee ee c.236 we(r+ fete +fx) ©.235 nether 1 “$s -4 7.135 71 1 =. 1.135 24 SUMMATION OF SERIES Seesno :141(26)+aedteat mat 11 1 171 1 11 1~3x8at)*tanqe~at)~353(qs~9) 141 1+aa(e-m) te (127)}loghn+logh(n+1)+...-+logh (m~1) 1 ly. 1 111 +ploshm—ao(5,—5)+sar(m~mw) 1 1 1~an(ns~8)t> S Qn+1 (128)>[rtews(33*7)-H} (129)x304F484... (30)1+9+ROHR +Feeey? +Fett... co 21SOAD ey xxsxt if: 1) (3n P+ +H+...©Grcomvergent) (132)f 1—222-ay+AB-—Bt xt =205—By +... 0 s a3yt1+20x+@apfa,(5+x)+524}+SPa,(5+2) 1 +OFfag(5+3)pape. 1ForvaluesofBy),s6€No.(146) EXPONENTIAL AND LOGARITHMIC SERIES 25 Reference =logh™wheremandnintegers X.141 =mlogh™ —mloghwheremandnintegers X.141 =3(1—logh2) A526 1=Trploeh&+VEE) where|x|<1 A197 ae ¥.456 Ball tx =Spleen 2+ F 1Z.165 = AE.12 ane eal AE.14 ee 26 SUMMATION OF SERIES SeriesNo. — san1 2, (l).* (sat5+xB.(5)+786(5)+F8(5)+.-- 1 i(3x) 1(3x)3 1 C355+3eB,(3)+SPa,(3)+SPa,(3)+... 1 i 136t5+6x(2)+GPa,(f)+...0 V Tot (37)x—(145)2+(14545)0 -(14}+549) 44.00 1x 13x8 (38)xFT +GG Hw LelyE134,113-54 (139)logh2+55?—3a" +rae8*ee© 2 (4)1-5--Hoe fda Bo2xs24a7 (4nx45-55 +FGF-...0 (a2)x320+F454... 0 xt2xt2-4x6 (43)FF +FS bo xP2dxtx5x6xTx8 (Mm) ~x4 4 4S -S-SPeB ew x?2x33x4 (49) 54 4Ft... 0 ‘tForvaluesofBy(x),seeNo.(1146). EXPONENTIAL AND LOGARITHMIC SERIES 27 Reference 1"S71 AE.30 - 1 AE.35Treva 7 a AE.43Teepe erg ate " logh(1+x) -Tex R.425 =logh(x+V+) where-1.<x<1 L.78 =logh(1 +V+x)wherex2<1 7.123 -—+- A190logh—— =VIFx?logh{x+VI+x} A191 1 =Tipe losh&+VIF} wherex<1 A.197 1 “=5flogh(x+V4xP L7 =logh(I—x+x2) L179 =yt low —2)wherex<1 F.197 28 SUMMATION OF SERIES SertesNo.xxtxt(146)ogh2+3+B- x od xt cantae —nF-wa nF ~Bas—DE... «0 3 xs (148y¢2[8,02—1)-B,@*-DF+BOS~NF—..-0o} x ele s(9)xtFttw (SI)tlogh2+logh3+...logh(v~1)+5ogh» (This series isnotconvergent.) x,Bt Bt Bs* (92)$1-5+Spt—Gpat+paste. xot 1 19 (153)145 38+ageTAGte© xtxt,x aneSede (514+5-FGtGt©1+Diary syeeeSaco etprsquoi! x x xt (56)5=Pa +aan © 4ForBy,Ba,ete.seeNo.(1129).+tForvaluesofB;andB;°,etc.,seeNo.(1129). q EXPONENTIAL AND LOGARITHMIC SERIES 29 Reference =logh (1+&) H.498 -sh N.1543 afc! N.1544eal : =14Stoghct—9) F338 =fe-cjorjer) wherej=Vai F.338 ee(1X,=floshQe)+nloghn=m GN~+B IP.612 Tan can t+ Groen . x -=4 22.123 x*tosh+3)ue ws =btantx+Hogh(+32) “x 2 =logh (I+3) ¥.80 =WeehC+DFwhere,Psisthesumofallproducts katatime, nofthefirstrnaturalnumbers Y.80 30 SUMMATION OF SERIES SeriesNo.x_ x x xt axSxe (5s) x4544 (159)Reversion ofSeries. Y=x—bx?—bx}—byxt—...cocanbecome Fayt + GP +Gy 4... if a CQ cy cor Cs G Gy See Van Orstrand (Phil. Mag. 19:366.1910) forco- ficientsuptoCp. 1 1 1 CO)1se ~TGF TET Bi34 G6)1+ ++ Gt O 11 1 (182)5+Sataete© 11 1 (183)+oatHste© 7 EXPONENTIAL AND LOGARITHMIC SERIES 31 Reference -logh<> Y.107 =logh(I+x+22+8+x4) Y.107 T.116 =b =by+2b2 =bs+Sbibz +5b)? =bg+Gbybs +3bz? +21672 +1464 =bs+Nbybg +Babs) +28(bs%s +Biba?) +84043, +42b,5 =bg+A(2bibs +Ahaby +b32) +12(3b;2b4 +Obyb2b3 +b2° +60(2b;%b3 +3bi%a?) +330bi%b2 +132H)¢ =by+bibs +babs +byba) +45(b,2bs +bybs? +ba%y +2bybob,) +165(b,%b, +bb, +3b,2b2bs) +495(bi4bs +2by%b2) +1287b,%b2 +4296,7 =logh{i+Ae F197oe e=Seca=BK F197zoMM=2loghn—"fogh(n+1)=logh(a—1) F197 =logh"| C368 32 SUMMATION OF SERIES x5 I 251 (164)—3+543?—9+Sgt © IX.Binomials, SeealsoNo.(1102). (165)30+marta+ODtatobat (166)1tet eee. © 5 5:7 5-7-9 167)2+55+grayt+Grgyte© 1113113-51 (68)1-55 +7aR- Tae t© 33:5 |3-5-7 (6)1+54+53+Tete © 113, 13-5, 13.5.7 (190)4pxFheFSLESTyy 113, 13-54 13.5.7 a7)1-5x-54* trae trae @ (The above twoseries areuseful informing certain trigonometrical series.) Tyg l4ne 47s(172)gx+FGBa+FETT +. (173) l=x+2-8 +...0 (174)1=2x+332=433+...2 1b ba3 113-5 (175)1+5x-5q* +7g6"~Taeeet te2 113, 13-5 41:3-5-7 (176)1=5x+7aFe+7ggg © | BINOMIALS 33 Reference =logh{logh(I+3") Y.107 =(@tay 1=Vi-u +H.468 =3y3 F.167 =v7 F.168 =v8 F168 =JZ+x4x T+ x? —(Wie =x T+x? ae +3) ©9=3y8 =(l+x)7 T.1I7 =(1+x)2 TT =VvIFx T.uI7 -__ TATVi¢x 34 SUMMATION OFSERIES SeriesNo.112 2S 12-587D)1453-55" +595" -Tee te Lobtby|147-10 (178)1=5PEtPEsgPEat.2 3.34 3d S113 (179)15xpt—FE TEEat 3359 357 (180)1=5x4$98FS+wo 1,13 7 1 (8)145-58 +geyt © 1,3 15,,195 (182)1=3x+3537—ag+apagat+© 1,2 64 (183)14bx-atthex-Death. wo 1,3, Uy, 4 (81-4x430-My Meee 1,524SE9935, (185)1+Bx-538+BwRatt 142 91 1729 (186)1=x4FatDe+TEat... (x).mn—3)(x\? cis1+n(7)+ G@) nn—4)(n—5)(x) +Mae Di)+... 2 =2 (2—(m2— (188)14a24OERa4OP=PP ay, nn(n?—12) nn?—12)(n?—32) eet EE ste j BINOMIALS 35 Reference =(+y8 T.u7 =(+ay4 T.uT7 =(+39? TT =(1437 TT =(+ xe T.1I7 =( 4a" TT =(1+ x5 T.18 =(+ayus T.118 =(+ x)"6 T.18 =(+ xe T.18 =z+VIA} wherex2<1andisanyrealnumber TL =(+VI+xwherex?<1 T.8 36 SUMMATION OF SERIES Series No, (189)Cy=Me=Don—2).sm=n+1) (190) Dt Cr Fee mC AoemE 191)mynmgCrmnt+mgC2mCn2+mCn (192) V+ mCy +mC2 +o++mEnHee (193) 1=Cr +Croes(Duy Hoo (194)Deggy+QemO2X oesMeCAME +oe (195) mim—1)4=DH=D), 4lon=Daa r+, (196)1+mMOHDggmontDm tr) 197)afx+p tMDerytee 4Mn=r=hn—Lap(n+Dgeayirg...} (198) 2n(x2 +y2y12 fortaMeat 4PEMD arty PREaMa D2) strty 4,} (199) 1=mCi +C2 = (= nC (200),C;—5s+Fas. if BINOMIALS 7 Reference C.186 =(l+x" C.186 =mtmCn C.189 =2 wherem>—1 C191 =0.where mispositive C191 =ml +xt c.197 =m(m —12-2 wherem«1 C.200 1!=AD! whererisapositiveinteger c.200 =(e+VIE+ +(x—VIFPH}whereisapositiveinteger C.204 =(x+VEEP (x—VIFF yA} wherenisapositive integer C.205 =(=1'_1C,_ where nisapositive integer c.210 | —yal 1eltptet 212 38 SUMMATION OF SERIES Series No. X.Simple Inverse Products (201)t Tofind thesum ofntermsofaseries,eachtermof which iscomposed ofthereciprocal oftheproduct ofr factors inarithmetical progression, thefistfactors ofthe several terms being inthesame arithmetical progression: Write down the nth term, strike offafactor from the beginning, divide bythenumber offactors sodiminished, and bythecommon difference change thesign andadda constant. 1 L 0)Taya +Tag tm terms 3 4 5 (203)3-4+Tag+Tyeteterms 1 iit (208)htgy+ggtemters te @ tliid 205)5+5gt5ateeterms 206)gh+gd+egBtoomterms (207)reatregteenterms (208)7g+gy+aigtemterms ratagt rote" $ 1 +=La ND +n some cases themhterm can bypartial fractions beresolved into the standard form when this rule can app. SIMPLE INVERSE PRODUCTS 39 Reference F316 1 1 “18HatDatert3) Fon aa rsTT) 3 nt+3 Ant+Aa+3) Hat n+ 2Wn+3) = F.322 n+l =I 2 “el . a! “aed 2 17___6nt+2In+17 ~ 36a+Ne+Yn+3) “nei =!—o« Ys,%6)'= F322 “3 (4a, Gey = . 40 SUMMATION OFSERIES SeriesNo.111 00)33+57+Fwtoterms +. re or 14-7*&7-10 ©7-10-13 “*** $e 4 5 6 QU)p33t+rat pagteterms te@ 1 2 3 Q12)5955+ost TetMterms: + 1 3 5 Q13)p33+aatastmterms $e@ 3.1 41 $1 14)Fy5trat Faytemterms 22Ea Q1S)x44+Fgh+gh++.mterms 1 2 3 (216)33+ps+egteterms 2.22 32 cory4252B43Ba.mterms as12+P4Ste. si 71 91 ai)SohtehSekt...ntems i SIMPLE INVERSE PRODUCTS at Reference 1 1 “127Rin+GnF3) 1-6 F.322 -1-__1__ Git Herts 1 -4 F.322 -5- mts “47 at DEF 5=3 F.322 11,2“6>n+3G+3rd 1=} F.322 =3-24 1a7ned*Mares Dd 3=i F.322 14 -1-s F.333 na=14 2 sper 5 F338 1a 1 “27273-57... Q@+) F338 att=!-aem F.333 9 =} F332 Laar-L 3311a BS 42 SUMMATION OF SERIES SeriesNo.20)F4BoeSaytmterms 11ul L O20T3+ratast mmeD7 S 1 a2) S—___1__ ¢)aaa ee (23)——tgBFDEFD *THEHDEFD 2! +EFDEFDEFIETHT “ 1 ye 0%ETHTAD y 1x xe O23)7+gas twarinary t 2aa+l),,oa+IXe+2) 26)§+HEED*+REET te 1 11 13001 OO)eatTinea*TaHTs 13-5 1 +pee © 4,aa+l),oa+INa+2) 228)145+56+tOeETDto? | SIMPLE INVERSE PRODUCTS a 14 1 Reference =3-23711@ F331 3nt+Sn -"eee 2h wa o—_——_. (+afl +m+ Lx} "a Ce, eye(Cg = A242)x¢l ot =oe -23whereaispositive T.118 b=1){(=1)*logh(1—x)- ~6-9(02)){orwpa=aq-are kee etkct 1b-a-l at] ted,oorrn)>=}whereaandbare positive anda<5;(4ii}etc.,arebinomialco- efficientsT.1s 2-4-6...2n -35-7...Qn+ 1) 12.267 =k whereb-1>a>0 AmB -a-l1 44 SUMMATION OF SERIES SeriesNo.@,24a+1),30a+IXa+2) 5+ S41 tbe OHH t? 1 1 O20)ED tEEDETD 12 +30 FEF DEED? 1 1 ©TERMED) *TFBMTBaEES 1 a 02)operant Pama teterms XI.OtherInverseProducts leat tod 233) 5+gt Zt © ity t, tyeoftrata tgytw fey 1 1(QO)35+sasteegte © | Jia*mas+567 ayy l 1 ©723eastKea yt 1 1 wo . 9T3353taro toate ' 1 1 OMataatsgte © 1 1 1 CY)Tag sat ep © Ly piv ay (2aiy(5)+(a) +(ra33)$0 | OTHER INVERSE PRODUCTS 45 Reference =aay tere2>a0 Aa -1 -3 n “TED Fae 1 1 a-rala-iA) Fae 1 =3 ‘T.143 -35 0,9224670337 E.158 Lai)-1=ron raBEBE02B619436126. 282 =logh2~$=ogsi471e08 F.338 =0.153426=4—togh2)©1SS42E HOA?oyare, =-g 0,0714ASH09O =B 6,033 -¥-3 O,0S4BESTYVSO —cas wr 39 “4-16 0,02990110(00 on ' 46 SUMMATION OF SERIES SerlesNo.11 1 9Taxa t5678+SOT ts? 1 1 1 C8)Tas+orrisas +9. t® ee a rsd” 556+ 67e $5 3,9 7,5 WU oy5-Stet B-Etmterms 24,8 6. 4)53—5-7*77~Feat*ShesMtr 4Hot totcot... 0 THTTRTERT 1 1-3 1.3.5 8)a+ra6t Daeg tmterms 111 OM)saatrs6t TET? 1 1 1 C9TratasetTagte 0s4Gt+otetit ® )aaa+ore+Toa + 252)et tt bw C2)Taxa +F567+e910 t 253)t+ +5) Ea 053)33+567+OTT+ 3135 (25)1-54 3TR 1 oe “1-3 1-1-3-539143 -24+heeraeate OTHER INVERSE PRODUCTS. 47 Reference =}rogh2-% 0,0424871013 0.31 “MErVvD ©0,004SB440817Y 0.371 =le@-9 0,098BT8/634O 4199 =3-2c F.339 _m-8 O,1168502753“6 ’ 1 1-3-5...Qa +1)37>SeeGr F333 =}—togh2 O,O56Q62781940 TT.44 -i(a-™) emanege EM ~F-pom2 —Codeiasyaiso TH =F(I+335)~floan3O,OWNME3 SYYO1.148 =}logh2 ONTILBOTIEL ~—c.252 1 “vA =Vy2 48 SUMMATION OF SERIES SeriesNo.G29LeStghytggtyg tn O515+ethgtapgteterms 258)1454934 34 26)1+HeteHPS+... | osetgitmate © 09yeghy+tglgtagto 267)1-354 FhFEE 08)STD 1 OTHER INVERSE PRODUCTS 49 1 ‘Reference =3 C.225 =-se} Aca =3 Y.505 =3 A197 =5 A.184 -#h11,2091 99576 ¥.505 =2logh2 1,BBELAYSEIS 1Z.136 =5(J-torn)ONGHS 16909, 1Z.164 =Fd+vd =©AMBOE ANY 12.165 “5-H? @YIETARTEG Sze =Be+logh@+2V2)O,2167432Y4E 812.165 =Jalentye A197 50 SUMMATION OFSERIES SeriesNo $ 3 08)>aD EIED : 1 1 13 0m)$+santret an) +sag tsaemitCO") Tong +57 +TORTI +© 1 1 1Opa teat Rat. © 1 1 1 OB) 1-s+ sR-ZRt i nL, (13), (4352|em1+(3)'+(su)+(GG)+(fees)+\ wa 12(1)?|1-2-3 (1)3|75)1+5(3)+$3(3)+339) +--2 : 1-2-3 C114ytPERtTpeat @ LiyeL121)?|11-2-31p |em1+33(3)+553() tars7 3(3)+ i 336 |3-69| 07)143426+PET+ i1 x x i 9) r73+ 34Taste 1xx 8)3+ Tas teat? 1 x x C8)(a3 Fast eT ® f | i i OTHER INVERSE PRODUCTS St Reference =! __ AS2 FEF D 7 “2 ALSula, 46s =3(1-3) D.495 =" c.372 =a H.476 12) _4 ~Tear” = A1% -35 1.236 =3 L.237 -= 1.237 -5 L237 =p etGogh A,wherewt<t T.195 1fl+x Lt yx=defSeounEME+20h-9-2}where O<x<1 T.125 =Lf=togh1+)-15Stantx}whered<x<1 Bve ‘T.125 2 SUMMATION OF SERIES SeriesNoXI.SimpleFactorials (282)+ftStte.terms 283)B+4Bye 1,2 3! 8)+Get Gat? on 2! 89)it Gai twa It? Sm! 1 S(<1yala -1a-2)...@—n) 1 can> nes) (288)Sn ¥ 1 (289)deve aadc 7 tlm id jm1 00) Timed tO mee 1,5.) m@tn-1 Onata Gra XIIL.OtherPowerSeries(Bernoulli's andEuler’sNumbers) r z 32 nt (292) [rent weet eet teal 1 1 1 1 ouitaestaaat teal ; OTHER POWER SERIES 53 Reference 1 n+l “2°@F27I F333 =Se4+2 F,334 on F.338 “@=Da-D! . 1 “a= eo ACG -Tend“Te+d a9 —P@a@ +1)=Teta 020 =(@4+!-1 =0.7965996-fite x (m— 1)! ~@F+Det2...@ +m)\ cai 1 1 -2- GFDa)yor4 =Jlogh2 1Z.324 =logh 2 12.326 34 SUMMATION OFSERIES (294)Ltfeetatet+tal 1 1 1 9[eatventvee1lama (296)(t=m4@n=my,Bn5!me 4@=ee) | 0)ut[Toes +eSae 245] newLV2a%n—1Vaan 1 2a —1. ‘ 1 099)ut[ata +amet 2 2 4 (onsaatwatt © oo . \ \ \2);YarYarVe*VeotViuptYerorVsteot>\ +Oe)| Yaszgnt+ Vunzepnot Vongzaot B+ 1 OTHER POWER SERIES 55 Reference -j 1Z.326 =3 1Z.326 =35 1Z.326 =2 1Z.326 =v=o 1Z.326 at,evel 1Z.326 =2 12.354 = own 1=Ay wherejal> A.66 a1 A.52 56 SUMMATION OFSERIES SeriesNo,(303)(n+5)logn—n+Sogh(2m)+235~saars2.200 2 2 Tin~360n3~*** (204)miViren{1+55—sastohy-..e12 360n’ 1260n5 *** nnlarge ‘Ontheseries Nos. (305) to(330), seeNo. (1130) for values ofa, f,ete.; seeNo.(330) forgeneral note covering Nos. (305) through (329). 111 GO)14++RtGt© 111 00) 1-4 E-Ft... 1 1 1 1 Cont EtEthtpt. 1111 Got1-f+E-Lep-.. 0 MaltiGOO)L4+E+EtEtRtEt 11111 GI)1-F+g-ZtE-Et @ 11111 OW1+R-p-gthe poo tm=3sn=09159656...i=Ooeiog435.2.‘Atalonrom38018decalpacesisgivenbyGlaisher,MesenerofMathematics, 42;49,1913. ”Mes OTHER POWER SERIES s7 Reference =logh (n!) G.140 anevim(2) G.140 2n)B,=SqS,=WOR AC.42 =wh AC.42 sySo Fah na, =UnUn=Zoe AC.42 n\n("6 AC.42 tytes =So +7) 4 =GyGy=sg AC.42 ; 2etf, =beV3ga001=oe AC.42 u, HhoV3aans=SE AC.42 7.140 538 SUMMATION OFSERIES ‘omt-f-deded baw OD1+e+ettt pte OM)1-£4+h- tieht OI)1+8-H Gettp 01)1-5-Rtet CIN14+E-Rate tw GI)1p-Ett GI)1+etate (20)1-f+geoget© (2)1shi+gig~wit+© CDlth tet m-Ete 02)1+ght-ga-ga te 0)1-gantgin—te | OTHER POWER SERIES 59 =WyWay=GES AC.42 =hyVIhagry=SMe AC.42 =ePot=ake AC.42 =tyVing=SEPT AC.42 =PuPaes=AERPs AC.42 =eTan=age AC.42 =anB, AE.3 -For(1-aa) AE.26 =owe Bost()) AE.30 =saoEe AE.30 60 SUMMATION OFSERIES Series No, 11ui1 9)5agatgmRTO - 3 -k 11ey 1 026)1+sot~Sami—aasi+Gats 11 1 1 G27)1—ja- gata tom o I 1 1 G28)1+goat—at—Tmt 1oai4 (329)1Ret TEt © (330) General note onNos. (305) through (329): (a)The values ofBernoulli's and Euler’s numbers are given inNos, (1129) and(1131). (b)Thevalues ofB,(x) andA,(x), etc., aregiven inNos, (1134) to(1146). (©)Thecoefficients $,etc.,inNos.(305)to(312)aregiven ‘inNo.(1130). Seealso No. (1101). (d)Thevalues ofp21, GamFant, ANdfz,aregiven inthe table opposite forvalues ofn=0to4. (©)Thesummation ofNo. (305) to16places ofdecimals isgiven inNo. (1133). (£)Some ofthese series arederived from No.(546), etc., giving @appropriate values. (g)Between Nos. (305) to(318) and(319) to(329) there issome duplication, astheseries arecollected from different sources, buttheresults arecompatible. ! OTHER POWER SERIES 61 Reference =GUO (jytBe “a Bay3)+¢nozs AE.30 =Pum AE. 59 =a AE. 64 =mt AE. 69 =by AE. 14 °le] -[=| _ 1 3x3 a Tn 2v228 VEE oe B33 21998 Ln 5-6175 2304vim|sam |ae|mane 3|307? 19x68|61-5477|4126sy2a|TSy2am |23.5 |Waesys 4 246118 11-13-1801%8WES TVd B.S 7y3 Ref: AE.63 AE.66=AE.73.—AE.77 62 SUMMATION OFSERIES SeriesNo,(oDesiQnyent ay4a) onOr {Bon()BariBoot(@) (=11Qn) i 1 i 39)Sari{*%(—)~254n(3)} (=IQn 51 i (333)Fa {Bn)+omBaos(2) 1 1~prtPi(3)} (=r tn 1) 1 i (39Sarit (4o(2)~a4(()} Intheabove fourseries scealsoNo.(1146) foran amplification ofthecoefficients AnyByetc, SlG35)a 1 1 1 TE BF grt @ i} 11I| Wow tee Generally 14 Se The dash sign indicates that only those values ofn (greater thanp)which areprime to2-3-4...p occur in thesummation BH1+kthehe. ria it (337)1Tate pte 111 (338)1tetatipt.. ~o | OTHER POWER SERIES 6 1 Reference =aha AE.59 1 =5% AE.64 1 =pra AE.73 3-Ba AE.74 =Us)2, 3,5...p—are prime numbers inorder =Us —2) =Kyl -21 ~3%) =Usyl —2)...1 =ps) Q.272 2 -F E154 n “2 E.158 z “> 27P. 281 vo Emeydopecke Becta: weneed 64 SUMMATION OFSERIESSeriesNo.Cle HtHt. © CM1+R-R-Ht to ONI-f+H-At.w . OO1+RHE At. 0 OW1-P-Rtht yo OH)I-PtH-Bt C6)Statint ptmtat? ONR-ktp-Ht ew 0)gtgtwate© G4)+t? Leoesterest, 60)2-2435-F4...0 @51)G-sstsipt @) OTHER POWER SERIES 6s Reference 2 -2 E155 _yz-2 A.364 2-2 Y. 50Ey « %A ok-% Qe E155 Pa “3 E154 Mxty2~1536 Ae Sas -& Y.501 =~ v3) ="a2v9 A.528 %—logh2=7—tosh ===6 “Ss 27, 2s-3 F332 = 4H.462 66 SUMMATION OFSERIES SeriesNo.tion G3)1-5 +-Bt e . 1G91-P+h-Ht.w 1 GH1-f+h-f+..0 05)1-EtH-Ht.w Sdoo$4 \,6s1454+b+...mterms. ptt. yv aaJ09Dae (360)3+3+3%+...cocanbesummedinfivecasesonly: @x=1, 53 @x=-1, 53 1 OTHER POWER SERIES 67 4Reference =5 T.144 . =o T.144 16 =F TL144 25 =% ‘T.144 oa a, by “6 n+l @FDG+2” G+ DOF DOD whereb,=EF! T.27 2k-7--@ Gy GF iat GF e+ Dass)” whereK=1.2020569=5°ysee(1133),D aC oeoe and¢,=£5(+545 +--¢44) T.27 = -Aft, 1.=1.0787 ila tarp 1 1 AC.63 *aaa tarea t)(otisher 1675 =logh2 2 “6 2 “12 68 SUMMATION OF SERIES Series no. . @x=2sinFZ,Ss ox(rmgy SE (61)Ss5 (363)s(1+3y 0)Sato (365)345 =floghxlogh(1+x)dx 5 06)3petty Dorr . 7S(ye ct) (ea]Sval 2 ' OTHER POWER SERIES. 69 Reference mil 1?=T-2(03) -3-(tosh2sini a a\?-5-logh(2sina) 22.286 -s Zombie deBalSi=6S,=2$)=5e HaeSs=S2eSo=W3e S,=877e $,=4140 A197 =5}Mlogh2 A.520 _ dae) toe antyo1 +aT 1 =3 A.52 ea (2 _ a =Deve f'Ztogh ede=2-2t0gha-FA.496 =z} “472 =! “4 1 1 . =Flogh2-4 _3 as 70 SUMMATION OF SERIES Series No, om> onSah + eet(72)2,aes 86°)>artyi St s6%)>aDi Sd N619)>oD >» 2.1Noa) Qeomite -¥4$x—js...0 "Spa 1 6)>lat arn 1 1 1 GP)eTtae iptep ts 0)1-24ME) _etee «+. ton +1terms (8p5+yex243Pt... o | OTHER POWER SERIES n Reference waefeS42+tosh9}wherex<1 a4 \C.246 “47” + -ae wherejx]<1 Q59 ~qewhere|x|>1 an #8 32-392 Sa BeBe 4-2 “ -s-2ERP= 8a =2logh2-1 7.142 3=3osh3-1) 7.142XS=-3+}logh3+2logh2Ae Ta= =(l+x - Y.107 “3-1 2/24 E.190 =t-dtd- 0 F197 =0. F.338 1=HTSpI+He+9) : n SUMMATION OF SERIES SeriesNo.Swe j (382) at (8)>aro F wee 1 8)2GOFF ly 2 3\rsyGB)’+Ey+GY’+2terms(5+5)+(G+RY+.mterms Sf13-5...20=1)?1 (385)>{24-6...20 }har $y=2logh2 -fe, | $=fey-1 | 5-2-5 1 1 |$=7Qn+D-3 wot | B=Z-} 1 1| $5=ype(IBu+13)— =J%1 |50=2255 6 | 13)=Faye(SO+43)—4 > 1 CMa 1 1 1 = oe =0.9159656 —D> CATALAN When ris anegative integer thevalue ofn=r/2istobe excluded inthe summation. OTHER POWER SERIES B ‘Reference Cad 1 =Gm =1~3m—1 on =Gm C.373 1 a4.74 -@) “5. T.142* sya1-2 1 11 $.=flogh2+5-7;2or+) 1_ 10 S3-37 oF 9 uid $4=35logh2+355-Fy(1802+13) 1178 $5"3~25"25 7 L $6=joglogh2+556~Tage(O02+43) : "4 SUMMATION OF SERIES SeierNe C89)n=3456.a 089>Ag 7 (8)DAT > L 389)aorAGT S 4n+1 (390)>4ED S An+1 CDZO Ha 392)>(-4adn + 7 Sys 4ntl G9)>AnGa in sD ee(394)>wo >pe (395)>a Saal96)>= o Lar c S le=1 398)>aaa j OTHER POWER SERIES 1S omReference =Behi@ Di T.143 =1-F =37-1 =logh(I+/2)-1 -i 2 =2_! “a2 =2-1 =i_4 “2-2 =-& wheremisanintegerandn=momitted. A.67 =Giawherenisevenandn=momitted A.67 =1 7.143 1=3 ‘T.141 1 32=2logh2 4123 7.143 21292 Y/ pin)364 teem 76 SUMMATION OF SERIES SeriesNo.111 G99)1=55+5aGe So 1 1 1 0)Lape lrtagtagto] Sc(401)x= o2}tx+abs)+$Ba)+FBMa)+...0 ’(403)¢1+a(x-3)+@PAy(x)+FAs)+...2 11,np,_ma-IXn-D py (408) 5+ Fat—De tw ‘This series may beused toevaluate B,* byputting n= 2,4,6,etc. ly2x 12 13 2x\ 93(t53) +2a(tea) tree(re a)t? 406)>5,57 = » (407)DOOM ST tr- 2-2 (408)14mx+BEMPT94PO2)a rly, 1.3) 40)1+(3)+32(F4)t@ ; 1, 12, 1/24)? 2 G10)1451+33)+353)+0\ ‘tForvaluesofAn(x)andBp(x),seeNo.(1146). OTHER POWER SERIES n 4 Reference =3 T.144 =2 Hu “Bi =Z_s “2-16 ol ‘ sete=SEZwherexisapositiveinteger AE.6 “25 AE.20 =1 Y.109 =x wherel>x>-I “YY.459 a4wherex>1 =fre+824 Mr+)where$=13+23433+... ‘A.197 =0 wheres,=194294394... A.197 =(e+ VIF. Y.107 wg 1Z,360 = 12,360 8 SUMMATION OF SERIES SeriesNo. at1pay 1 1 is aineh+H(5)+(53)H xa+x)_ a+xP 4a+ x) (iyEt_HOT, ete, (Qn—2)!x(a+xp +(1pAEE4. 13)1=ax~9+EP2g—9 ~=He9)a—ay4... 7 aaLimaor Ta+x") a ed| (1)LinDae { A x x xs G16)Lim(25-PatSate «) XIV.Trigonometrical Summations (417)sin@+sin28+sin30+tomterms (418)cos6+cos20+cos38+tonterms (419)sin8+sin30+sin50+...+ sinQn—198 (420)cos8+cos38+cos58+...+cos(2n—1) (421)cos@+sin36+cos58+... +sin(4n—1)0 | TRIGONOMETRICAL SUMMATIONS 9 Reference =a-3 1Z. 360 =x A.199 =(1=x%whereJal—9]<} A199 1 =3logh2 A.201 ~l 1a A.201 andears A.201 _ol nd 8=sin5(n+1)8sin“>cosee5 E,283 1 nd e=Cos5(n+1)8sin=>cosee5 E,283 =sin? n8.cosec 0 E.283, =5sin2n8-cosec 0 E.287 =sin2n0(cos2nd+sin2nd}x(cos8+sin}cosec20E.288 | 80 SUMMATION OF SERIES SeriesNo.(422)cosee#+cosec28+cosec4#+...mterms (423)cos§+cos28+cos72+...mterms (424) 5cos3+...mterms Soret med (425)1—2088+3.c0s20—400s30+...mterms (426)3sin6+5sin26+7sin30+...terms act 427)Dksinks (428)“Skcosko (429) >(1)coske T (430)SsinQk—19 (431)"S(—1sinQk—9 + *) 2n (432)tan6+tan(+2)+tan(0+22)+...mterms ! ‘TRIGONOMETRICAL SUMMATIONS 81 e =cot5—cot2-19 E.125 1 «|3nb 30=00843x—190sin3cosee3? E.287 1 =3 E.288 =058+(=DMn +Deos(n=DO+nc08 mBayy ~AI+cos . _sin8+(2n+3)sinnd—Qn+1)sin(n+18 E17 = (I=cosBy 7 (2n—*) ancos(—.—]@ =n (7 Te 4sin?3 2sin3 «(2m —1 _ain(zD1coun Te2sing ‘int 1 cof No =-$+ (pe 7.82 20085 sin? n®- T.sin8 2 i 2) ne=~noot(5F+nf) EB 82 SUMMATION OF SERIES Series No. (433)cot+cor(@+2)+cot(0+22)+...mterms at2nk? (434)2cos one(435)>»sin=e a! (436)>,sin2 (437) >sin?ke (438) >cos?kei 5 11 l | (9are+aaattST> \ 1 1 1OOartmatt ET> 11 1 |ayaint+ant30+--+Sint—Dd i i 1 1 1 '42)Sag*sarae+--+Sat=Dp (443)2conec?(=) 2m (444)cosec!0+cone?(0+22)teemterms 445)tan?6+tan?(0+2)+...mterms | } TRIGONOMETRICAL SUMMATIONS, 83 Reference =neotnd En avn PFsin8) =Bi(I+cos3+sin) 7.83 =u 2 sin™)=E(t+cos—sin) 7.83 =cot5, T.83 <n_cos(n +1)0sinnd“3 Zein6 7.82 #2, cos(n+1?sinndtin 7.82 =Rt wheremisodd A210 2=EE wheremiseven A210 m1=> wherenisodd A.211 nt=>whereniseven A.211 =$02~1)wheremisodd A223 =n®cosec?n@ wherenisodd =J2cosec?whereniseven E.73 =micot:(3+nd)+min=1) E73 ! 84 SUMMATION OF SERIES Series No, (446)sin?9+sin?20+sin?30+...nterms (447) cos?6+cos?28+cos?3+...mterms (448) sint6+sin¢20+sint30+...mterms (449) cost8+cost20+cost30+...mterms = aim+(= (450)>tant(2) 2n 4n (ne 2274cop paar (451) cot? +cot?+...+coA dn 4n (n=l)e rd ‘e @=De (452) cot*5+cott5"+...+cots 453) 1+200s8+a?cos26+...4atcos(n—1)8 (454) 5atcosko $ Nos. (453) and (454) areequal. (455) >@*sinko | TRIGONOMETRICAL SUMMATIONS 85 Reference3.1 an™cosee®—Lein3 in3"conne28 =Jing(n+sin"cosecs—jsin3(n+1)8sinScosee E.288, =feosin+1)psin™®cosee$ 44.083(n+1)0sin3cosee E,285 =5Ln—4005(n+1)8sinn8cosec8 +cos2(n+1)8sin2n8cosec28) E.288 =4Ln+40s(n+1)8sinn8cosec@ +cos2(n+1)sin2n8cosec28] E.288 =fn—Dor+n3)wherenisodd +A.223, =fm- De-2) wherenisodd 0.349 =gp—Din—240?+3—13)wherenisodd0.349 _1=acos8+attcos(n—18—arcosnd ~ 1—2acos6+a (1=a0#1—arcosnf)+a*ttsin@sinnd TeT= 2acos0+ a . __sin(1—a608nf)—(1—a.cosarsinnd 12 1=2acos 6+a . | 86 SUMMATION OF SERIES SeriesNo.(456)asin8+2a?sin20+3a?sin30+...mterms (457)cos?0—3cos?30+35608?320+3com?30+...t0 terms (438)cos©+cos2+...+co2%=De S189)Dting, S(toe8P won>(aes) 461)>(2rsineZ) (1 U (462)>(gstan*5.) 5 at - 1463)¥——>(©>ae) eto oS > (465)sin@+sin(0+f)+sin(@+28)+...mterms (466)cos0+cos(0+A)+cos(0+28)+...mterms (467)sin8—sin(0+8)+sin(@+28)—...mterms (468)cos8—cos(6+8)+cos(6+28)—...2nterms ! TRIGONOMETRICAL SUMMATIONS 87 Reference =[asin @—asin@—(n+ Narsin(n+1)8 +2+Dae'?sinnd—(n+Dart?sin(w—18 +nar? sin(n+2)8—2naesin(n+18 +na**4sinnO\/(1—2acos@+a2)? ‘D.502 1 ne=f{30o80+ (-3)cos30} E.126 =-1 =Pooty—2cot29 7.83 1 e\2 =eosec?#—(5,cosee=.) T.82 =(2sin3)’—sina 1.82 2a —1 18=Fre+4cot?20-55cot5 7.83 =1 ere mi =Tea" where misoad A218 =},whererisoddandmiseven A218 s 1 in"8 B =sin{0+5(7~np}sincosee E.282 =008{0+5~1y8}sin8cosee§ E,283 =sin{a+2548+mhsinEFseo8 E.285 ; 1)9) «i=sin{0+(n~3)sinnfseo E,288 88 SUMMATION OFSERIES SeriesNo. (469) sin0-sin20+sin28-sin30+...mterms (470)cos6-sin20+sin20-cos38+...2nterms (G71) sin 0-sin30+sin28-sin40+...mterms (472)cos8sinB+cos30sin28+cos50sin38+...mterms 473)Yrsin($ +78) T (474) —6sin(a+6)—SsinQa+b)-...+(x—7)sin(na+b) (475)tsin8+asin(0+f)+asin(6+2B)+...mterms (476)tan0tan(6+B)+tan(@+A)tan(0+28)+...mterms (ATT)cosec8cosec20+cosec20cosec30+...mterms (478)sec0sec20+sec20sec30+...mterms (479)>sin@+kB) + sin0—asin(@-) Sumtoinfty=S288 —Dwerea<1 : TRIGONOMETRICAL SUMMATIONS 89 Reference =llr+1)sin29—sin20m+18)cosee@ E288 =}sin20m+1)0-sin2ndcosec@ E.288 =Joos20—}cos(n+3)9-sinn8-cosec8 E,288 =sin{ro+500+19g}sin(20+6)cosee$(20+6) 1. .=jin{no—Fon+198}sin320—fycosee5(20—8)E.286 _(n+I)sin(G+nf)—sing~nsin(¢+7¥1)-I=038) sin(na+6-25-*) =(2Dsin(ra+b)+(9-9 __sin{na+b~(a+=}_Tsin(a+6)~6sinbasinaye (Gainaf _Sin8—asin(@—8)—a*sin(8+nf)+a**#sin(0+(n—1)8} -T= tacosp+a E17 tan(0+nf)—tan@—ntanB =Beeta—ne—tans E.124 =coseeB{cot@—cot(n+1)8} E125 =coseeoftan(n+1)6—tan0} E125 =sin(2+)sin($4)pcoseo§ 7.82 90 SUMMATION OFSERIES Series No. (480)2cos(6+kB) 2(481)1-BoPamesaN? —3ins9tees +(EID2Qsind8 (482)sin@—PVgins94MOP—VM—3)sins... +(=A sine 8 (482)1-—PTFcost94=UM?=3coseg212(n2—32Y(n2— —(B=BNE==9s4.+(D2 eosO1 (482) ncos@—2cos +(12402 cosBy? (482c) mcos@—me cos +(-10-1 cose 8 (4824)1—Frcost+MOH2)cone 2 2y(n2 ——Me =Pcosso+...(—1p?21coed Hi TRIGONOMETRICAL SUMMATIONS CF Reference =cos(0+)sin("$+)posseS 7.82 =SMe wheremisodd ‘A.204 =sin where nisodd A.205 sinnd ; E.64 =1vaSOwheremisodd {io =(197 28wheremiseven veSoa : j 67 =(—1)2cosndwheremisodd fests n i E.68 =(—1)? cosné whereniseven {ote 92 SUMMATION OF SERIES SeriesNo. 2—29(483)sing—MP2)sins ind— m2— 4Me=Me?~4)ins4. 31 +(=12H sinOye 2 2-23aay1%sinra +MO? sine+... +(=1221 sine8 ns! (485) >{cot(6+ra)+cot(8—ra)} _ = z (486) >{cosec? (@+ra)+cosec? (6—ra)} 7 sin@2sin202sin276 (87)Feosd—1 +Feos28-1*Teosz—1 *°*terms (488)tan-t2ytant¢tanthytantty 3 7 B ae 1aoe +anTas (489)2>——— 4Txt—axcos+a? (490)2>;——___4+—_Tx3—Daxcos2+at {Tissummaton anbeated1int,heumbeingcorm,wheethevalueofmisunrestricted, and0=5. , TRIGONOMETRICAL SUMMATIONS 93 Reference =ie wheremiseven A.205 =cos6 sever 7 =cosné whereniseven A.204 =ncotnd—cot@wherena== A217 =n?cosec?nd—cosec?@wherena== A.217 2sin200 sin8 *Freos2 +1ZoosO+1 D.330 ae =tant E126 1=~~ theremiseven Y.55 wow y-a ¥ta et 1 j =PE Aywerenisoad Y.55 A.207 94 SUMMATION OFSERIES Seis A @=) 491)Lim[Hfeee$+sec2?+...weOO] S tan”Qn (492)2tant(1252 sinh6) (493)(2.0084-1—(n—292cosBy» +POD Ecosyr54 sect (-Depe set (<1(n.6056) (494) (2.0088~n(2cosoy-?+N=3)2cosOyr-44 soot (-1)2n 6050(nisodd)ve12 (niseven) (495) 1+1+...mterms0s0+cos36*cos30+cos56T**"term (496)im[2{sinaZ+sine34.site3 (497)Lin[1—xcos6+x4cos28—x9cos38+... co] (498)Lim[xsin6—x4sin20+29sin30—...0} pan (499)Sesinno (500)5acosné7 (501)cos8+aos38+a?cos50+...arcos(2n+10+ 1 TRIGONOMETRICAL SUMMATIONS 95 \ ;Reference =jloehtan(5+3)wherew>0>5 12.355 =tantRee whereiseven A528 =sinn@-cosec6wherenisodd E61 =sinnO-cosec @where niseven ~ =2cos nd E.63 =Jeoseetan(n+1)?—tan6} E125 =carast whereisapositiveinteger 12,326 =} A.276 =fran A.276 asin®=oe wheatct 7.139 =ee whereat<1 139 1—a)cos8 =aa wll< ams 96 SUMMATION OFSERIES SerlesNo.(s02)sano+tan(2+22)+tan(0+2)+...tan(0+2) (503)0058+50s20+3.0830+...«0 (504)cos0—$60820+$60838+...00 (605)0080+3.030+$0854... (506)cos8—50s30+feos50+...ey (507)00520+}c0s40+$c0s68+...00 (508)sin0+5sin26+$sin304...co (£08)sin#—}sin20+3sin30.+... c0 (510)sin6+Jsin36+$sin504...0 (S11)sin@~$sin30+$sin58—...o0 (512)sin26+5sin40+4sin60+...00 1 TRIGONOMETRICAL SUMMATIONS 97 Reference =Stan56 E.191 38 =—logh2sinwhere0<@<2 A.356 =logh2cos§where—7<0<a A.356 1 6=zloghcot5where0<0<3 A.356 =Flogh(cot$wherex<0<20 A.356 2 2) ~ =]4where—5<<5 A.359 =—logh(2sin6)where0<0<= A356 =}0-9 where0<6<29 A.356 6=5where —7<0<9 A.356 =Fwhere0<0<9 A.356 =~Jwheren<0<2n A.356 =Floghec@+tan8)where3<0<3 A389 =}-2)whered<0<n A.356 98 SUMMATION OF SERIES SeriesNo.(513)cos+be0s¥F+$e0s¥F+Feast+...oo (514)tan0—}tan?8+gtan’6—...00 (515)5sin6—2sin26+Fhsin30—Asinad+...0 ($16)cos8+300828+4.00830+...c0 (517)c0s0—3500820+460830—...00 (518)c080+450830+4,00850+...00 (519)sin—sin30+3sin50—...«0 >sin?nd on2 (621)sind+}sin20+Jsin30+...00 no 4sindaadein (52)sin§—Lsind04 Lsin$04... 00 (523)sin?0—}sin?20+3sin?38—...00 (524)sin20+5sin30+fin4?+...00 TRIGONOMETRICAL SUMMATIONS 99 Reference -i{2V3logh(2+V3)—=} E.123 =6whereZ>@>—@ E.107 sinh6 =Fsinh= Ua =4-2 —hetwhere0<0.<20 A.360 n=F where—2<0<a A.360 =J20whered<0<7 A.360 =FOr+2)where <0<0 A.360 1 ca ®= where-F<6<5 A.360 ca 7 3n=F-9wheret<a<F A.360 =5Hr—8)where0<< Q.163 =emia)—where—7<0<m Q.163 4sin@35 ELT 8“=F JUL =jloghsee@ ©.334 , 2%) 1=~sin@ogh (4sin?5)+5(wr~8)cos0 100 SUMMATION OFSERIES SeriesNo.(628)sind+4,sin20+4sin30+...00 (626)sin~3,sin20+3sin30~...«0 nodei Lg (527)sin8+35sin38+55sin58+...c (528)cos0—4,0830+4,0858—... (529)0080+3.0820+3.00830+...c0 (630)c0s0—3.60820+3,00830~...o0 ($31)e080+00830+200850+...00 (6532)sind—Hsin30+sin50—...«0 moa dei 1. (633)sin+4,sin20+fsin30+...00 SFcosineoo &sinno(535)Lane 2 (636)a0080+$c0s20+$00838+...00 (537)acos8+©cos30+S0858+...00 : TRIGONOMETRICAL SUMMATIONS 101 Reference =h@—9)~20+w)where0<0<2n A.362 =OP—O)wherencO<am A362 =5(20—wi)where0.<0< 1 7 7=g(78+082)where—5<8<5 A.362 ion "eget~ar(g"-*) where—5<0<5 A.362 1 Tt=gg(2040—=)©—24)—Tywhere<8<2xA.363 1 7={0-278 +eo}where—7<0< A.363 =554—rb?+=}where0<0<7 A.363 7 7 ®=7852—467 -f s =(Gn?~462)where-F<0<5 A.363 Ca a = etep where0<0<2m 7.138 =—Tee=Dwhere0<0<2 A.370 sinar =Tsing=9where<8<2 A.370 sina =—flogh(1-220s6+a)wherea<Land@42nw LoonLt24 2 30+ 6=jloeh TS where|a|<1 E122 102 SUMMATION OFSERIES SeriesNo.(538)logha2~2(Leos0+515.0820+tscos30+...<0) (538)acos0—$00830+%c0858—... (540)asind+¥sin20+©sin30+...00 (641)(~~(sine+shysin29+stssin30+...oo) (542)asind—©sin20+©sin30...co (543)asind+$sin30+Sin50+...0 (544)asin—©sin30+sin58—...o0 (545)10050+Ao0s20+400830+...0 <=i (546)>sinan—8 = jan-t: 647)>(91EAcosan~198 ' (548)2asin6+4a?sin20+6asin30+... <0 <cos(n+$8oT >sin(n+$8my Sets, oesin(n-08 60 ’ TRIGONOMETRICAL SUMMATIONS 103 Reference =logh(1—2acos@+a2)wherea?>1 22.302 =Jtant226082where[al<1 E122 =tant280 wherea<1 2Z,302 T=acosd 1 asin8 =tan-t2507, whereat>1 22.302 -1__asin@ =tantSeay where lal<1, @#Qn+ Ie E122 1sgy-1228108 =5tantFYwhere[al<1,6mr E122 —liggn Lt2asin8+a? =flogh+Festewhere[al<1 E122 cos—1 =erg Whereat>1 22.302 =4tan-t22.6082wherea<1 T.140 __2a(1 =a?)sin8=Tap Where<1 22.903 =loghcot§where0<<a A392 =5where0<0<2 A392 = where0<0<2n A371 104 SUMMATION OFSERIES SeriesNo. (652)¥Ses—0 7 (553)amaemt 7 >£0816 555)er 655>(pens (558)a>(—1yrtSere r oynSum (60)25"sinndQF+a (561)5(yeni S cosnd (562)Nees! S(=I)feos2a—1)8_cosAn+1 (663)2Sart -areata : TRIGONOMETRICAL SUMMATIONS 105 Reference =—mcotma where0<6<29 A.371 =b+28—6)sinawhere0<0<2andmis4positiveinteger(omitm=n)A.371 =(7—8cosmo—88where0<0<2xandmisa positive integer (omit m=n) A.371 asinadn—8) is i =ER where <0<wandmisodd(aunrestricted)A371 =Teesaty=)where<6<mandmisodd(aunrestricted) 4cosfra A.371 sina?=FERS where<2<0<9 Quist moosad 1=sinna~Dawhere—7<0<a Q.191 _meosha(n—6)1 (22 =—Snhaa awhere0<8<27 AE.5 _sinhof—0) (Aan =Thine=9where0<0<20 ori =sinha here—<0<mn 22.117 sinhwa cosha6L =+33nhoa Jawhere—7<0<m9 22.717 ad 1=142~29—20526—20sin20—}60s40 106 SUMMATION OFSERIES Series No. «2ndsin2078 (6) GS SeeNo.(572) 2nn8=,cos228 ea (69)5a SeeNo.(573)(566)3sind+3sin$+sim44...2 a 3 pte (667)sin?6+22sint§+24sint9+26sint$4... 1 1 1 (568)cos28—3cos49+3cos60—1c0s86 +3608108+...c0 1ly 5 (569)1—jtan?20+§tan*20—5tan®26+...00 (570)1~jtan?20+jgtan'20—757tant20+...© (51)1—Fram20+SSEtant20—...«0 >sin2nn8 EmrDeas >£082nd (573)t2>= Gr2>Came Ssin2nd (1t2>Gap S£082nd (752DGm +For 4x(0)seeNo.(1128).BeareBernoulli:numbers. | TRIGONOMETRICAL SUMMATIONS 107 Reference =}a-6 whered <0<a 1 aw~(j0-9) -%whereo<o<a 3 =7@ C.336 =R C.336 =logh2+loghcos@where—7<@<a 22,302 2tan@ =jan O.383 _2sin=ne O.383 4sing-as 0.383 =l_g@-=1- (A.370 37% =5-6whered<9<1 fey at al 1 (A.370TO+B)=3("642)whereo<o<1 (AIM =oy, h=DHA wherek>Lando<e<1 4.370 -Gye 1=GruO+(-1B)where<0<1A370 108 SUMMATION OFSERIES SeriesNo. (516)tsin200+SS+ES+...co cos4r8|cos610 (57Htcos240+SEHEH4S0864og S 267)>tan \ ry (579)>(=tant2 48 138 50 ($80)tantgy—tantge+tant 1 oi a 1 6 Lraeea (582)see?+greet+Leeth+...wo m4) a wa) (683){F-Spsino-Fain20+{3-A)sinze ~Fsindd+...0 ‘n>6n) ‘m3Gr) (584)-$}sin0—F-3}sin20 (m36m) +{F-Gpsinse— co 1 2 .3 a (585)5(1+@)sin8+(1~©)sin20+31+@)sin36 10 +Ad-e)sindo20 fd m m4 (300){F~ja}cos—Feoos28+{5-A}conse ~Froosso+{F—Shcoss0—...00 +tForvaluesofBy(8),seeNo.(1142). 1 TRIGONOMETRICAL SUMMATIONS 109 Reference Daagentt=(“ISTBaet(®)where0<#<1 AE.3 2ee-igde=0(B+ (IZ}whereo<¢<1 AE.3 7 tanh 7=a7-tanae A.314 71sinh78 gtenaed asa sinh7 =tan!a A.314 1“ =57cot@ Y.54 1 =cosee?@ —J Y.54 7 “> U.4i a “> U4 ze : =f u.a Ca 7 =&-ole U.45 110 SUMMATION OF SERIES SeresNo,(587)0s0—}&cos30+£c#cos50—se0 QnDEY saefeosan+1)32 (588) > oe 689)>tll +(—Drte)sin 5 (590)1+a?+4acos6+2a%3—a)cos20 +2a%(4—2a)cos36+...+2ar(n(l=a)+(1+a2)cosnd+... (591)2+1+dacos0+32Dcos20+... 4Ane=1)+@+V)cosnd+...00o <et (592)Daeapltt —@)+(1+a%)]sine (593)>Heapgttat—1)+(at+Winn Memorandum: Consult Edwards Integral Coleus, Vol. II,forextension tothese series. (594)esind5Orsini sw (695)<Peone,PecosFyMO eo (6596)§sin20+&singo+...4ECMnsinnd+...00 TRIGONOMETRICAL SUMMATIONS mt Reference ==}tan-tcosBcosecha) E123 =a—6 where8spositive, |]<a 5.309=a+6wheredisnegative, |@|<a =Jetwhere0<0<a UL42 ___d-ayp=a papWherea<1 22.303 = (@—1p =e paywhee>1 2Z.304 sind =Ga whee<1 2Z.305 . sin 8 ase where>1 2Z.305 in8 j=tant5a5wheregispositive 2Z.194 ==logh VI= 2c COST FAF wheregispositive22.194 =ASS" where0<0<= t i wM2 SUMMATION OF SERIES SeriesNo.(597)$1+coshm)sin+3(1~cosh»)sin20 +fp+coshn)sin30+...oo l+e . Al—e). wa -C98)areaintoyagein +OSsin00 = (= 1(9)2G cot S__c0s2nd 0)>oars 6Ssin$n-I)n-sinQn—198 on1eS S905(2n+28 0>See sin(Qn+28 9aT (04)ae+teo >a 5 .= $ nm cos28cos40 <.608Fcosnd 06)TySet == j TRIGONOMETRICAL SUMMATIONS 13 Reference cosh@ =rome ua =whereo<0<a U.S 2=(G-) were-acoca 1.139 =}-Jsine whereo<o<% 7.139 3 1=358where0<0<3m =2wherety coca AB.243 2 3 3* 32 =F-8)whereFn<0x =cos26~(5~6)sin20+sin?@togh4sin?)where 0<6<a T.139 =sin20—(—28)sin?6—sin8c0s6logh(4sin?6)where<O<" T.139 =jsin#~sinOogh(2sin3)where0<0<27 —A.368 atatcose—1ir— sind where0<0<2e A368244 2° 7 " 7 1 =Foose— 14 SUMMATION OF SERIES SeriesNo. 2,cos™nsinnd 2sin284sin40 2 conTt cos20cos30 S(160sno om)SFSpath SE 1 60820 cos.4_cos60 (9)5-TS3SSTTe (610)44Fcos@— 1cos29—<4cos30—1cos 40—...c0 2°54 13 24 35 aa we,sin?cosnd cos38|cos50 sin08 (yar+gt = 2.> 10 sin3@_sin50 sinSin | 19)IeRete=Da | a is sin?sinnd sin30,sin$8 z (13)aptet = int | 300s38Scos50 =sinFncosnd (19pe tO (615)22252?436834co=>Boomez S(=D608nd (616)>wtetd (617)5sin0+53sin26+£38in30+...00 qT ‘TRIGONOMETRICAL SUMMATIONS Ms Reference =Fsing dot tou=3—40088—50sind =Fsino 7.139 =horOsind whered<0<x Q166 =forO)sind wheren<0.<2 =~fortsingwhere—7<0<0 =bcos Zsino+dosing =} Find +3sin 1 1=5000s@—Zsin8 =4sino—4sin6togh(sin6) 4 2 =~40sind+cos ==~feos@~cos#logh(2sin$) =(€088+60528)ogh(2.005)+asin20+sin8)~cos8 Q.190 aa=(2sin$)"7sin2? E116 i 116 SUMMATION OF SERIES SeriesNo.(618)1+5cos0+$3c0s20+...00 sin26|sin36 _< sinnd 19)F53tPagte©a=eaTD 1cos36,1-3cos50 (620) 080+5S FE teoo . 1. 13 (621)sin@+5sin30+5]-sin50+... « (622)1+$0820—54cos40+513,00860—...o0 sind_sin38_sin50 >_sinnd (oySEES s7 Dates sind@2sint@_2-4sine6 28)VST .2. 24. 2-4-6 (625)sin0+3sin?6+53sins0+FFsin?O4+...co cos@|cos20|cos30 6a3+taatzast 1@ cos38cos58cos70 ©)Ty3- p57t+eT Isint@|1-3sin4@ |1-3-5sina B77+244+246 6te (29)1+Faint0+F4sinea424Ssins+...© 2 2. 2.42.62nigFsina+EAsinea+AEsins4...2 (630)sin+acos@—$sin8—$cos#+...c0 .3)1+ac0s@+$00820+$00538+red 1 TRIGONOMETRICAL SUMMATIONS 117 Reference O12 0=(2sin5)"cos*>* E.116 3 1 1 =78nd —5(7—8)+5(n—Acose =feos"!(I=2sin6) c.334 L_ g(t 42=yeaain(F+3) E17 =Veos (I+e088) where-F<0<F E.118 =gsin?@ wherenisodd X.54 =Gcot Y.86 =0sec Y.505 bcos—Leos20 c.421 ; =}00s8—jos . * 1 . 7 =Zoos?—foswhere-3<9<5 A.369 o=loghsec?3 1H.498 PY -= L.78 =sin@+@) Y.84 =e608cos(asin) x2 118. SUMMATION OFSERIES SeriesNo.(632)asin#+$sin20+$sin30-4...00 2(633)1+SOG, ew. @sin2#|atsin4d <a2"cos2nd 92an S$a4cosQn+1) 393 SatettsinQn+1)8 n> Qn+ Tt Sasin2nd 638)>oar -Osi@cos20 |276sin3@ 3304cos40 (639)1—asing+SSR4,BOsae_Breede wo @sin2022630304si (640)0.080+ESn2F_Rhcos3?Posingsw (641)rosin+SFsin29+Ssin39.+...2 (=vee, (642)1+sing+C+1)sin?e+HDsinte4+26sin?e $2eine, oO (643)1+27200850 +222cos27S+220852S+...00 = nm OSeahG (644)6088+00830+=Dcos58+...00 1 TRIGONOMETRICAL SUMMATIONS 119. Reference =excessin(asin0) X.2 $605(asinOferso+eso} E118 =Jsin(asinBylercore—ervey E18 =cosh(acos8)cos(asin#)wherea?<1 1127 =sinh (acos6)cos(asin 6)where a<1 Ta =cosh (acos8)sin(asin 6)where a?<1 Ta =sinh (a.cos6)sin(asin&)wherea?<1 Ta =cos8where[o|<4 A312 =sino $=tan-t2)—sin0 E131a e =<5 L79 =#0080 Y.84 =2cos"cos(n+1)8 X.22 120 SUMMATION OF SERIES Series No, (645)sin8+nsin30+"=Dsins0+...00 lia aa ~2) (646)5+25c0820+AGT? cona0+...co (1)costo—MD cose?asin?a +MD =D)cose40sint6—...zero (648) ncose 9sin8—MW"—2)coge-ssin?0+...2er0 (649) cosnd+neos(n—20+=D cos(n~499+... If misodd there isaneven number ofterms sothat the last term contains cos 8. (650) cosnd—ncos(n—2)0+mYcosin=40+... +peein)GD} (651)sinnd—nsin(n—2)0+mYsinon4+... +omraemet) Intheseries Nos. (652) through (659) ,C,arethebinomial coefficients. C=Mba Wn=Detn= r+) (652)082nd+2,C;608(2n—2)8+2,CsC08(2m—4)0+...1 +5ul, (653)608Qn+1)?+aqs4Cy608Qn—1)8+...+aqtiCy6088 j TRIGONOMETRICAL SUMMATIONS 121 Reference =2cosesin(n+1)8 x72 a.\2 Qe{r(5+Di=a 8% where aispositive butnot necessarily aninteger and—z<o<5Q.263 =cosnd E.33 =sin nd E.33 =21 cost E.55 =27-1(— 1)? sine0whereniseven E57 =27-1(— 1)? sine @where nisodd E58 =21 cost @where mis even C.278 =Ba costar! C.278 12 SUMMATION OF SERIES SeriesNo(654)cos2n0—2,C,6082n~2)0+2,C,08(2n~4)0+... FED bac, (655) sin(2n+1)8~ays1Cysin2n—1)8 +aer1CzsinQn=30+...+(—Drapes,sin (656)1+,C,.60828+,C3.0s48+C300860+...+60s2n8(657),C,sin28+,C,sin40+,Cysin60+...+sin2nd(658)1~,C,0528+,C,608.48—,C;£0860+...—cos2ndACiSin20—Cysin48+,Cysin68—...—sin2n8 (659)Cysin28—,Cysin48+,C,sin66—...+sin2n8 1=,C;c0s20+,C,0s40—,C;60s69+...+cos2nd (60)1—atsin66—%°=)62c05200 — 3p+YFossin300+...2 1m Qn=Hen3) (661)5+FScon20+OrnDcos40+... (662)sin?0+2.5sinto+24.4sino+...20 11s 1s9 (663)cos0~460830+4.5cos6—4-52cos+...c0 whereh=4f°”cose0dah (664) General case of(663) =n l=n3-n cos#=300530+=5co50 l-n3-n5—n PPR teas+...00 When misodd, this series terminates. TRIGONOMETRICAL SUMMATIONS 123 Reference =221(—1)sin c.278 =226(—1)sin2st9 c.278 =2rcosn@cos"@wherencanbeoddoreven =2+cosr@ sinn6_ where ncan beodd oreven =(—D-0?2 sine 8sinnd where nisodd =(12/22 sine 8sinnd where miseven =(=1-072" sin8cosnd where nis odd =(=1)"?2" sint @cosn@_ where nis even =cosaf Y.SIL =Pee |cos?»-16| wheremisanintegerQ.191 =e Q.130 =feos 8 (3wisld)=1.113 x.36 7*rz(i) r(g+=Leoswherec=4"costa=$2.G+)(+3) wheremisanyvalue X.63 124 SUMMATION OFSERIES SeriesNo.421).(665)sine9+5-3(1+32)sin60+... 4-6. ..(2n—2)2) 1 1+e BfgetGapsini120 2oy 2 oy (666)1+$5sinto+57sint8 22.42.62+PT sins +... 2 (6615cost#—5(1+3)cost2 ata (pa hal costp— +5(1+5+5)costo~...co 11i111 (6612)5008203(1+5)oos30+3(1 +3+})eos40 —...c0 sin® 1 1) sin?@ 660)Se+P(b+) 1 1 1) sins6 +2.2.8(5+h+3)or+200 252 “ian (669)2>Gepga(1—Ae008nn)sin et1|2kStecosnm—1 (670)—*aheoacosnd 2rn+O Fx=aos248 (671)FT in 0)?(672)tan-x+(isin)sino—OS®sin20+...00 tan?@2tant@ 2-4tans (673)1tas tap teo ' TRIGONOMETRICAL SUMMATIONS 125 Reference =o .335 2) Lp=ea) 7 1 2=j[tosh20s?5]where—7<0<7 ©2582 1Viogh4cost)’—© ° A.529 =§[locos?5]'-5where—7<6<7 . a L.82 ~in 7 =dt where0 <0 <9 =dt where <0< 7 __x(x" —ac088)=Hin=Dearcosd+a ¥.55 =tan (x+A) where x=cot Y.108 =2a8 Y.505 sin 126 SUMMATION OF SERIES SeriesNo6741-"™Y tanv +MOOR 2H=Dane. co San?2-2 (15)>{anroan3 ¥ Cretan 2 -2 (677)0scosa+5.6082cos2a+$c0s38c083a+...co (678)c08860sa—$60828cos2a+5.60830c0s3a~...00 (679)sin8sina+5sin20sin2a+$sin38sin2a+...00 (680)sin6sina—3sin20sin2a+3sin30sin3a—ew (681)c0s8c0s.a+3cos20c0s2a+35cos30c0s3a$e (682)c0s060sa—4560820.082a+3560830cos3a...<0 (683)sin8sina+3,sin20sin2a+3sin30sin3a+...00 TRIGONOMETRICAL SUMMATIONS 127 Reference cosnd=S288 wherenispositive integer D.330 =tant +tanBABYomity=x A.3I4 x tan ax =—tant+tantSMomity=x A314 - x sin wx. =—}logh[4(cos0—cosa)*]_where@#2m+aA358 =ilogh[A(cos@+cosa)?]where@4(2n+IwtaA388 _Vpoap Sin?HO+a)=floeh SE8where@#2mm&a A.358 gen S240 =) wy=plone 2)whereOxn+Deta4.358 =de4h@-abewhee <0<a A.361 atast@-m—dba whereas o<a A.361aera iz . =etHe4%where—G@—a)<0<(a)A361 itadeeb (a—a+(0—99%)where(x- a Latha —aP+(0—a}where(n=a)<0<tn+0)A.361 =}Me0)where-a<0<a A.362 =fal=0)wherea<<(n=a) A.362 128 SUMMATION OFSERIESSeriesNo.(684)sin8cos.a+}sin29.cos2a+}sin8c0s3a+...0 cos#sina+360s28sin2a+}.c0s38sin3a+...00 (685)cos0+ac0s(0+f)+a?c08(0+28)+...«0 (686)sin0+asin(0+8)+a2sin(@+28)+...© (687)cos(0+a)+5c0820+a)+400s340+a)+...o 042m ta (688)cos(0~a)+$08200~a)+460830-a)+...© 042m ta (689) sin(@+a)+5sin200+a)+4sin300+)+...00 (690)sin(0~a)+}sin20~a)+4sin30-0)-....0 . @ (691)sin6+asin(@+A)+sin(@+26)+...00 | TRIGONOMETRICAL SUMMATIONS 129 Reference =(@)wheref(6)=—38 1 ito<0<a 8@=5("— a) 1=8(9)wheref(@)=5("-8)1 ifa<0<m 2@=-40 1 I)=4(—2a)=g(@) A.358 __c08#—a.c0s(0—p) ="T=2acosB+a E13! __sin—asin(@~p)="Tacos +a? ET 1 eal=—}logh[ssind50+a A.358 1 al==}logh[ssin?5@—al A358 =}-@+a)whereO<a<a and0<(6+a)<2n A358 =}{-@-a)) whereO<a<m and0<(0~a)<2x; when @<a thesum isdiminished by A.358 =coorsin(@+asin) E.121 130 SUMMATION OFSERIESSeriesNo. 2(692)cos#+aos(+8)+F;c0s(0+28)+... cost cos! (693)1—c080cos8+2°cos28-SF"cos38+...00 (604)sin0—Jrsin(@+2)+Frsin(@+49)—... (695)c080—Hc0s(0+28)+eos(+48)~...2 o a (696)acos0—¥cos(0+29)+Zcos(0+48)+...<0 (697)>Asinnsinna608mB (698)sin8cos8+sin?@cos28+...«0 Sly 2 5)>52ndsin?ng (700)>sin?nosin?ms TRIGONOMETRICAL SUMMATIONS 131 Reference =©4¢08(0+asinA) E.121 =e+608c0$(c080sinB) E.121 =sin8cos(cos A)cosh (sinB)—cos8sin(cosA)sinh(sin) E121 =sin(cos8)cosh(sinB)cos —f) —c0s (cosf)sinh(sinf)sin(0—6)E122 =$008(@~pant22°88_sin(9—gytanh-1208iN E122 where 0<B<a<3 =5M—a)whered<0<(a~p) =je@+a-—-ha0 where(a~f)<0<(a+B) =fale-6)where(a+p)<0<a A.390 __sin8(cos0~sin8) ==a whereo422 E17 1 0<2<npTwren Ey A391 =140 A391 i . 132 SUMMATION OF SERIES SeriesNo. 701)>Jsindn0sin?ng (702)>Aisin’ndsin?ng (03)cos0sin@+5cos?0sin26+}cos?@sin30+...00 (704)sect0—Fran?0sec?0+$tant0s0020+...00 (705) cot6cosec?@—cot38cosec?30+cot50cosec?5@+...mterms 2 3 (706)6+cossin0+2258sin26+£958sin30+...00 sin@|1sin20.1sin30 (707) Os OR @ @cos8 (708)loghsind+acot#—Fcosecta+FASE —...0 12S1ym C708)+SD,pittyrcos2el (110)5+42"sinmncos2nd TRIGONOMETRICAL SUMMATIONS 133 Reference 1 =[n A.391 a1 A391 G =3-6 whered<0<5 E.121 D=sindcosd 22.246 =e A.23 “3 Y.108 “3 Lsi =logh sin@+a) Ln =(=1)ifBisnotamultiple of =0if8isamultiple integerof! Q.191 0isareal variable between 0and 1 nis anodd integer>35isthegreatest integer inn? =1whend<0<5and3 <0<1 =1when}<0<3 Quist 134 SUMMATION OF SERIES SeriesNo.XV.Hyperbolic Summations (711)1+cosho+282%5SOHN gy (12sinho+Sh22,sinh30©7 3 (713)1—3sinh?6+24sinho—...20 (714) 1+acosh8+a?cosh20+...attcosh(n—1)0 (715) asinh0+asinh20+...«0 1. ly (716) sinh0—3sinh20+Isinh30—...20 (717)sink?6—2sinh8—24sinh%—26sine%—...co 3 B Bo (718) tanh0+Jtanh?0+Ztanhs0+...<0 S41 D (is)cotnt0—>,tanh?5, 2 tanh02___eae(rot~2= nett, 2n 2tanho ER 2 sinh201ONGea2sorian | HYPERBOLIC SUMMATIONS 135 Reference =£26cosh(sinh8) E12 =ccosh sinh (sinh 0) E122 6 =inh0coat A198 —L4@cosh @~a"coshn+a**!cosh(n—1) E17-— T=2acosh? +a . ___asinho =T=2acoshP+a Buy e -$ E.122 =e €.336 =tand— Fant#4fans0—...cowhereBisbetween+7 E123 =gacoth?3g, c.315 =tanh n® where miseven 136 SUMMATION OF SERIES SeriesNo. __tno 1 nz nsin? (721)t|(tanh@+coth8)+2= tan? 7 2a=tanh6 —~ enimtT tanh0 On (722)¢ —— ns(yt +2 1seaetan?241,2n yatanh@—2— a nsin?= coth@ n (723ypSE. 2= tan?=7 vi! 1(01, >SaaT,- 2n inosinh *2'47h3 2tanh@Hse sion28 1+BORED~AED20~cor] tat ___2sinnaSNRe,Taq8”GohO=cosTR soho *3°93 ' HYPERBOLIC SUMMATIONS 137 Reference ‘=cothn@ where niseven =tanh nd where nisodd =coth n@ where nisodd cy i =tanh>wheremiseven 138 SUMMATION OF SERIES SeriesNo. jo12cotha+SS——+—_ oo 2=| anno +38nh5 13 ‘annyt)im3+>aT (726)t= ime ttewake +7tanhy oS 1 1Jooths + —=come22simsanetptnhy 1>cosnd (8)540D Snsinnda2 1 S(=1)"60snd 00 5+ S(etn sinnd (731)2)— XVI.Trigonometrical Expansions oo < 2" O33)1m =DOMGH, BOs < Gent 39)0-54 =DDBET : 1 _ sinh@ 1ERT, 8” aah=cosBR sino *2'0h3 “ : TRIGONOMETRICAL EXPANSIONS 139 Reference no j =coth™where miseven no ; =tanhzwhere nisodd no ; =coth>where nisodd ‘cosh a(n —6)="—Shan where0<8<20 A.393 =7Wh=where<0<2e A393 sinh aw cosh a@ =7Oat where <O<= A.368 =7Sinha’ where—2<0<9 A.368 sinh a@ =cos@ where 8< =sin@ where @< © 140 SUMMATION OFSERIES SeriesNo.8Os <gat030-45 =DMeT Inthefollowing series Nos. (735) through (763), seeNo.(1130)forvaluesofthecoefficients. e o Spoe B35)1=BFBay =1->Boa7 @rs <Gt-d 36)17FagHo=DTT ao So C3+RtBRto=DBaa (3)1+GtEt mL >Ene @ o <Om (9)2455+2G+P=24DKH 8ao6 Sark (740)Sgthgthgt-@=2Say eo Spoe CADhthythat. @=heDhag @o S$On (742)HotWS+Magytoo=HotDecay 8@ 6s Bet 08)ngtngtngte©=Dway y TRIGONOMETRICAL EXPANSIONS. 14 Reference =tan!@where@<1 E.107 8 oor8 =Scot AC.41 a =tan§ ACL =#cosec8 AC.41 =sec0 AC.41 30cos-— AC.41indosing sin§ =z AC.41 cos 3sin -—2 AC. AL 2sin2 3cos -—2 AC.41 20s 2 3sin$sino -— AC.41 2sin2 142 SUMMATION OF SERIES SeriesNo. 6,8, 8 Sgt 48)01+OtGH=D.EH @o <n (49)1agaR 1D oa @05 +am (46)T+HHTG#---=DTGHL eo <Bn O41)JoAG+agtoB=0+DIneo eo SgGn O5415+SHHB=+DSRT U@ os Ssgaat 08)nataatet ODME al Som (50)Ro+RG+Raget+©=RotD>Rey a o On O31)PotLG+Pagts=Pot>Pra ©. 6 Se ptet052)Q0+OF+Ogt+=DW_—TI 2 2OD1+BFE G+BGto.© oe (154) 0+Bax+Engte. Ss(22-1—122", (755)logh+>TamBe . 2 2 (756)By0?+Ba37,OF+By57a08+...00 1 TRIGONOMETRICAL EXPANSIONS 143 Reference; D fe sinBos5=-«— AC. 4 208% 2 _3cos 28 -ae AC.4I sin20 =D AC.41 3sin20Re AC.41 _£0820 “5 AC.4 _3sin2@ =e ACAI £0828 “3 AC.41 cos @ =a AC.4I sin@ 7] AC.41 =sec8+tand Y.500 =logh(sec@+tan0)=gd-10 Y.500 z=logh tan@where#2< 7.123 ==loghsng BL245 144 SUMMATION OFSERIESSeriesNo.S$ 402 S212"=1) C15¥tootSips}=~>EPam = a SRet (758)logh9+toeh{1~=ars}=tosh—Y=p20 T 7 B,®|ByO|By06 139)StPtBERt© + B,G2 (760)—SY281cos2%BxOh> 221Gn)! (61)5+$+ZO+...wo 1,S221—1 3+Gem Be 16@ 1S 27» 08)4g3~B= 5DBBO @|26 <22a(220—1) (163)047+ +...0 =SSS pp345 >Gayl 88. 6)2Grape ae? 2 2 2 2 08)<B- F50t —w wED Te? S@ (166)2> T(Bote a 0605+etre tetetiet 1 S28“37Dame 444 4 (88)9 eet ee eT? | TRIGONOMETRICAL EXPANSIONS 145 Reference 2 =loghcos9where @<= B.237 =logh sin@where 6?<n? B.237 =tosh, A315 2sin5 =loghS882=cos8 ALBIS =cosec@where2<n? T.121 =cot@ where&<x? T.12 =tanwhere#<= T.121 =tan where0#nz A.296 =tandwhereox2De c.360 =tand B.237 =cot@ where 0#ne C360 =secO+tanOwhere>0>—F Y.501 146 SUMMATION OFSERIES SeriesNo.1 1 1 1 1 (0)9+ 5-40 eat edt law >oeate 2 2 2 2 O”)TF ee -S( (Qn=In -d¢"EF 7" T a) 6 1 3 5 ”)anaamtTe © 1 L 1 0g-@—atALERT | 73) — +te te tn.| Gop Grp Gt GepT I\ 74H1+204204PyML Ve13 Sam | 19)0435445 +?=Dmaior| a|1408(116)1+5+EPte. | CT)1=84A20240 Stent OB)O45+ES5+ =DH 1 1:3@ (779)1+OtSBE0 op SEB2n=1) r4> Qn le nl 18120 04Sed & (780)0+554+555+wetsDasa a 1 1 TRIGONOMETRICAL EXPANSIONS 147 Reference =cosec@ where @4na C.361 =sec?where0xEt, €.361 sec 0 ir=a E,158 1+ @cosec 8 =< E.158 sec? 9 = E.159 =tan{3+9} H.498 =sin-!@=5—cos-!@where@<1 T.121 _0 =and H.498, ‘=cos? 6 H.498, =(sin 2 where |@|<| 7.12 vers“! 0 = Y.505 -1=Sesto Y.505 | 143 SUMMATION OF SERIES SeresNo.(81)-S5-egtgp 182)#4 183)0+F045RosPOH4.oo cs0-32 FAG a&-8+5-..0 (136)140-28 2PE ew (187)044RAR 2 (788)5+TH oo (789)+404 404...0 (9)0-F48-F4Fw (9)$4Borw. C9—306+Seg... 00 9Jo Fea. cw 795)0F4Bc (9)$+BeBtw +No.(792)hasbeenomitted because itduplicates aprevious series.—Ed. i TRIGONOMETRICAL EXPANSIONS 149 y 50 +30Reference @+t -19— += tan“!@“a 1Z.135 =sin?0 D.336 sin“) 6 =FSB wherea)< LB =Vi=@sin-t0 A191 =}+arto $e H.475 = cos 6 ‘H.497 =sing H.497 =logn22 1.80 =loghtan{F+6} H.498 =logh(1+sin6) H.498 =logn20*8 ¥.80 =logh(1 +@sin@) “YY.106 =logh #cot@ Y.80 =logh(1+tan) L79 =loghsec H.497 ] 150 SUMMATION OFSERIES SeresNo,cn40495 OFa 98)oft5+SFE+...oh (199)1445064O64... o0 Gon14045+a 014045 -F-ERE 603)1454Fa4. 004FT 4. 2-2Papa 009F=fgpa=FDpaye ea}Soon Faaaco 08)140+5-24 0)140454 aw C10entpet eet © i TRIGONOMETRICAL EXPANSIONS 151 Reference =ene LB =coo ‘T.126 =atin ¥.80 =logh cos@ ‘YY.107 =eine ‘T.126 =eoe Ly =ie+revcos¥3} E.190 =fo=fv{eos243—vain283} E.190 =tan-1@ where@2>1 T.122 =sec!=F—cosec-1@ where@>1 T.122 =ame 1.126 =euncte T.126 meme Y.107 =Ftan3} A225 132 SUMMATION OF SERIES SeriesNo. =h oh? (1=30) 1tan6+Pg~amSeate A 6Ae (12)sin0+AS+GeL422mptata dt..3 h 22-1 WR (13)506-10-+aS-ageape+© oS Any? (_@\e 64aDaremre) 1,¥_ 61S_20 6195+2oy 3tLoe4 7 1 1 1 1 C1)eq Seat 3 Ite 2mn—2—12)n2—29 ry1%PRED moe=136829 1ae Llow VToe 1 1(1+3)-F(1-3)+F(1+3)-F(i-g)+--- (819)z[a+D-@ OSH... +I+EE+. ae 1\6 11\6 emF-(145) 54(1454s) F-.@ 18 (a 1)@& wn55-4914}Ftit i 1 1, 1y\e+fetaltes)+g(teg esa Yo 1 $+ ntnDgmpatwap | TRIGONOMETRICAL EXPANSIONS 133 Reference =tant (0+A) L1 =sin @+) Lm =sec“! (0+ A) L1 =tan-10where@2< Ti =neot n(n=Oexcluded) A217 =7(%%-1) A225 =er C.421 To =cose+She A.389 =cos! 0 ¥.79 =}(ant6?where|x|<1 A191 (an-1 9=e Y.89 =1?cosectnf A218 154 SUMMATION OFSERIESSeriesNo.Yo 1Ss(62+3n2) 3)>gapet ea Yui 1 SO +662n2+nt 0)gaan te Wao Hap 25)ae +GE pt 29BtZtTet Ht © Y_a-e(27)20a 2n(Sn— (628)1OD ew 1 @ pert3 62)H{@~OF+...Raho —...oh 1 1 1 1 1 (3)b+gttetgtetatmat 2 1 1 1 1 (»-(Fy+a)-(ata) 2 18 1T\1-365 1-3-5/1 11\0 055+(n+pzas taaelnt Ets)Tt (9148ELEMat+I (34)042042484... 0 mn 2m 24 m2\2 (835)wert tate Heta)eee=} | ‘TRIGONOMETRICAL EXPANSIONS 155 Reference =wcotx0cosec?28 A.225 =wt{coseetx0—Jcose?wf} A.225 =tne 421 =cosec?@ A.222 ==(cot79—cotna) A225 ‘sin 8\"=(3) H.498 =sin? 0 Y.79 =Foot A.225 ® 8 =3tanzr A,225 4in 2 z =g(sin where 3<9<35 A.223 eet “vise se -aoe where[8]<1 A197 =tan A.196 m 156 SUMMATION OFSERIESSeriesNo._4B+|6.70501+oH (636)901+)—LAUEOD,6-7+6" _89-10 71+0) so UtOF5.0 on1MED, =Din=D3) (638)@—Fae +$+—nFae +406- NZBe+...0 (839)140+SF4HDG HOLD MOEMEMG S 1 0)>T= mm—1)(m—2) (841)mg—Mon—hr—2)gy mn =Wim=2Xm=3h=D5ay (42)1—MODon4,mem=Nom=2K—D544.c0 (64)1400+PTEon4WEI a +SEPMpcos(ntant2) +...2 mla (844)mg—MOE=VYgy4min?=WYP=)gso TRIGONOMETRICAL EXPANSIONS 137 : Reference =tan-1@ where (1+69)? < A199 =2"cos" wherenispositive AIL 2 =—logh cos@where#2< B.245 =tite L717 2oeect™+Ze®1 (A217 =Bn28?ns*Finims2 A.218 =sin(mtan OL+62)n72 Lal =cos(mtan-16)(1+62)? Lal =cosb? LB =sin(msin!) L.16 158 SUMMATION OF SERIES Series No. -me—ne—#)054...co (246)>{loan(1+2)-3 47)>(1p{tosh(0+‘)-3 4)+(1-545) E+(1-pe ede. (49)5,5—5,5+5.0—...0 Sunltgtptg 50)¥(-- (54,+3) ¥ 1 i 002rors} InSeries Nos. (852) to(854), seeNo.(1134) forvalues of A,{x) andB,(x). (652)2o{x—3)—2aye)4OMaya)—... (853)—2a)2By(x) +OOBa)-Caepax)He 654)1—apes+OMyay—OMgery+...2 . .md (855)>CO"Gor >ae SeeNo. (841). 1 TRIGONOMETRICAL EXPANSIONS 159 ‘Reference =cos(msin~! 6) L.76 =loghsin78—logh78wheren#0 A314 =loghtan%—togh2wherenx0 A314 lant 146=5tan-* Ologh7+5 A191 =}tant#togh(1+) A191 1 a=cose(x6)~5omitn=0 =ee- mmitn=0 A.25 inex =sinsy ~¥— 9 =asina@x—1) AE.25 sina =2008a(2x—1)—acosa AE.25ina =2008a(2x=1) AE.25sina =(1+ @y? sin(tant 6) L.81 160 SUMMATION OFSERIES Series No. 2 i620 (856)2CYang SeeNo. (842). 65)Dare 40)Daa S (<1) 09)>Sem TTH InSeries Nos. (860) to(862), seeNo.(1134) forvalues of Ae. (860)2a4y(x)-2ayn)+coeASS)+o (861)20{4,00)-24i(3)}-SPfase-24,(3)} +OH(ase)-xa45)}moe (882)capats)—224,(5)}—SY{aay~2444(3)} +BHade—284d5)}-.-- Inseries Nos. (863) to(873), seeNo.(330), etc.forvalues ofp,q,r,andt. (863) py+9:0 +pyc? +gua +... (864) ry+tha+rya?+teat+... 1 TRIGONOMETRICAL EXPANSIONS 161 Reference =(1+62)?cos(rtan-6)—1 Lat 7sinh7/2+sinx62 AaB =BYcoshO/2—cosx2 . =__sinh2ny ~yeoshZny=cosZax A314 7a 135 ~eGCosh9—cos8)~Zao Q 2a AE.93sina =#08Ox—Na Ago cos@ ee AE.93,cosa Stewcae AE.80a ocd_7a sn-3) -7—1 “6G=3) AE.80 6 4 162 SUMMATION OF SERIES Series No. (865) pia +20? +psa? +gaat +... (866)ra+toa+rya+tat+... (867) pya +psa? +psa +... (868) rya+ra? +rsa5 +... (869) tra? +tat +tga +... (870) gna? +gaat +qua? +... XVIL.Hyperbolic Expansions eo Stn 71)1tatat @"Loy @®6 gett 82)0+5+Rt-@- ae HYPERBOLIC EXPANSIONS 163 Referencea(t47 sin (7+22)itis 9) AE.$0 2cos2z a(t422)sin(3+-anat(5+3) ‘AE.805208% —1 3 za naOG -m— AE.81cos24z zacos32 =m6 AE.81 3200874 —1 3 sin2 za é -=—s_ AE.81 V3200872 13 sin7 xa 8G = AE. 81 Nie z =cosh@where#2<co E84 =sinh©where62<co Ee 164 SUMMATION OFSERIES SeriesNo.@|265(1767B73)O-34+FEBG © - 122a(22~1) =Dom 2g ae m2 meen?+12)men?+12m?+32 674)15EE eetT co ®os BI)OFFHZH © SS14ant= 81)DOOgreaETT = 2B) OD OBE1+DWAGRE=14Bastashe S (22-1—1)B,62" mt1+2> mS @ 706 3106 =1-$4+35-mt ? 1 1 1 1 (80)5-0(Sre-aree torre) 1 3 5 (804(sae sareetare) 1 1 1 1 65+(shat greetorpat) $=1-CM (89)9a S Be+31+(19 89) & +ForvaluesofBy,seeNo.(1129).No. (877) hasbeen omitted because itduplicates aprevious series.—Ed. ! HYPERBOLIC EXPANSIONS 165 Reference =tanh@where&<7 H.498 =cosh ma C421 =tanh-1 where [8]<1 H.475 =wy2cosh7¥3sech73 A.3I4 =Ocoth@ C.343 =Ocosech @where 62<a? c.343 =cosech@ Q.136 =sech@ Q136 =coth@ Q.136 =sinh? @ Y.80 =cosh} @ Y.80 166 SUMMATION OFSERIES SeriesNobb bgs)? bl ©)SSR” GarretGaeR~paee|)aT b b b ©amt arnt are +...0 -[eeejoeT @1406 81-Fe 8Os 8«-2(0-F 4-...) (889)14PEEsino4EHVME+nse4.co (690)Fsind+P29sins124212PCEes 14 Fw eos (892)9F-F—.. w 24|2408 (693) 1-7 cw 2282 2496 26910aieeeTs 20327052367 24 (095)0Te 220S|2367 89)6—SpSp+p++2 | HYPERBOLIC EXPANSIONS 167 Reference aft 7 =35/5-Tan A.501q a 7 1 a I-ifee-+] asl 6 =ahd H.498 =cos! (tanh logh8) L.80 =sohe L.812088 sinh@cos Lél =cosh (8cos@) L.80 =sinh(8cos8) L.80 [email protected]@ +H.497 [email protected]@ 7.127 [email protected]@ ‘H.497 =sinh @.cos@ +H.497 168 SUMMATION OF SERIES Series No. OS _Ot > Et (897)1-7+H—HGH matt CMaR For E,*seeNo, (1131). S (2n)! (898)>(-'sae a >Baan +1 S Qn!=logh20+>Gap >2m (699)logh20—>>=,CML_gan 2Bay trout=logh20—bai—$3gu++© S Qn 4, 900)3-1 Payee Ila 13d 67238tTas @20 2.406 0F428 2Ae Forvalues ofthecoefficients inNos. (902) through (938), seeNo. (1142). 3 (902)a(x-3)+042)+Ase)+...0 (003)(2ay23,cx) +2"Byay+POayn)+...00 (904)1+(2ap2datay +OMayn)+re) $e.0 (905)(2ap2,(5) +&ai)+... | HYPERBOLIC EXPANSIONS 169 Reference =sech@where@?<5 ‘T.127 where @<1 =sinh!@where@2>1 T.128 =cosh"!@where6?>1 T.128 =sinh-15=cosech-10where02>1 7.128 (sinhore ¥.90 =Guinhox—1) AE.19 sinh a =dosha(2x—1)—acosha AEG- sinh@ " =Zeosha(x=1) AE.19 sinh a 7 _=2asinhfasinh3a =Tae AE.33 | 170 SUMMATION OF SERIES “G00Lae2af)CHa) 4a3 a ONG, ar ONG, (007)1+@apAi(t) +2"Ad(3)+...2 (908)a—SPa3)-GSa3)ar) 00)30GPa) -Gals) -.-- (910)20— a,(t) -GYa,(2)-... ~SPB) -52G)} 17oS aft)GPa!) 013)Ho—4ap+“Bay —... (14)2-F{o(§) -a(})} -#fvaft) —ea{t)} -. @15)1+earA.(3)+oOa3)fees00 016$+@-G- DAF-@-Ne-DBF +@-G5-DBS+... 17)A-26- as+2433—aS —2535—DBAS+... ! HYPERBOLIC EXPANSIONS 7 Reference =—gsinhde AE.35 sinha a =TsinhFa AE.33, a “Gale ‘AE.32 _3asinh 4a=Tsinhia AE.51 3asinh 2a =Saha AE.43 _3acosh $a=Fcoshta AE.51 3asinha —Sinha AE.35 3cosha= E.492cosh3a A 2cosha=Cosh2a AE.60 @cosech=3cosech 3 AE.33 acosh 2a “Sink3a AE.AT acosha =Snh3a AE.47 172 SUMMATION OF SERIES Series No. 18)1+2ap4,(3) +on445)+oe45)tee 62)14BFBHBS... 022)p~flat+BatBas+... 923)(60)8,()+SHa(3)tee 024)5aSPa,(3)—GP(5)-... 025)3a~$,{orn)-3%8,(3)} 626)(1228,() +ea2)+owe#8)$e (9Jo2a!) 29)— (931)(2a28,(5) +Gas) $e ! HYPERBOLIC EXPANSIONS 173 Reference =acosech a AE. 30 =asecha AE. 32 1 =-atanhba AE.30 =}acoth5a AE.47 _(sinh 2a—sinh a) = Dinh3a AE.45 8asinh asinh 3a snide AE-33 __WOBasha AEST ___a=esha) AE.48 —12a sinh asinh Sa =sinh6a AB.46 sinhSa =3[eoaha+cosh3a+coshSa} AE.46 _asinhga=sinha AE.60 _3asinh 4a =Fsinha AE.48 —asinh da AE.60sinha __2asinhjasinh3a =sinha AE.40 174 SUMMATION OF SERIES SeriesNo.2(2a)i (934)ta?—tat+tea®—...1 (935)(2078,() +253) Hees0 (936)2o{x—3)+SPayy+BFago +...00 037)1+@ap2date)+SHAgcy+MFAgee)+...00 038)pra—py+psa—...0 2_>_@m! (939) logh>—YO)"_gm ,ras 2-18 =1-30=behg- 53-Ta ge (940)logh3+S(—0rsae,O 25 2,181.30=logh§+5-5-Fag © Soet 1tod ON)>Po=ptptt eaa (042)5cranny 1ForvaluesofpandseeNo.(330). 1 HYPERBOLIC EXPANSIONS 15 asi =~ Saha ABA _ _asinh $a=-agate AE.35 ah 72 _za Sh AE.75 V32coshF-1 __Jasinh 4asinh asinha AE.33 _asinh a(2x —1) =-——Shha AE,19 _acosh a(2x —1) -——Snha AE.19 za “4aad AE.60V2cosh2 =cosh-15=sech-!0where@<1 1.128 =sinh =cosech-! @where 8<1 7.128 =tanh!5=coth-!@where62>1 7.128 =! 29=Fsinh 6 ae 176 SUMMATION OF SERIES SeriesNo. (943) 3(—prevent $ 044)5(-pre 7 9Daron = 1 1 6>(oemare torre222 KAbotpttptpte /i 2 2 2 |O®\+roatietipet% {1 1 1 \OO)syntaeRtyte | 111 |00tie trpEt © 1St \O503+2pam < 1 02)>GTS oo(953)&-Tgpte© oP pO 26.06 =2BSar~BsTart Bseer a- (954)14MODoseo 2 2-4 (956)t6-50 +5365... tNo. (955) hasbeen omitted because itduplicates aprevious series.—Ed. HYPERBOLIC EXPANSIONS 17 Reference -—, 7.129*Teosh} 7 =}(tanh0—1) T.129 1 o=—jloghtanh5 T.129 _1 sinne 1 “2cohO—cosa4B Ald ° “lf *=rcoth=|=tHe) ¥.55 =Feoths Y.55 =Ftanh=» Y.55 =Frank5 Y.55 =Seotha Y.55 sinh 2ay=}coshny—cosInd Ald =JoghSih Y.109 =cosh"6 Y.80 sinh-1 6 “VIF e Y-90 178 SUMMATION OF SERIES Series No, XVIII Taylor’s and Maclaurin’s Theorem 057)$0)+O) +F4O... THO+... x x (958) $0)+24°)+5#'O+...HO+... ‘XIX. Bessel Functions Ss(a1yartr 59)2,Fae 2xt x6 | 060)1-354Fp parat © Hl x xs x?061)5argtarg~aerE tT© ' x x a | 00)5{\-araeptraepera— ij= i (=1yxet2| (963)2Fee+741) : Px,xtxt 09)[2{1-4-2} :(965)(2)"”cosx ! weve 66)Vian(1—x41 +VI—x8 XX. Elliptic Functions 067[i+(ye+(aye tees| ca ‘1?k2 1-3\2k+coos)51(3)T-(23)$- | ELLIPTIC FUNCTIONS 179 Reference =$a+x H.481 =4) H.480 =JyC2) Q.355 =ols) Q.355 =42) Q.355 =J) Q.355 : =J) when nisanygeneral value Q.359 2).=(2)sinx=Jy) Q.364 =Jyh) 0.364 xJ,(nx) when nislarge and0<x<1 Q.369 _pt do-fveran |where0<k<1 A190 2 _,-fVI=Bsin20d0 where0<k <1 A,190 i 180 SUMMATION OF SERIES SeriesNo.XXI.VariousIntegrals n-1n-3 la 08)nd FE Ifsin2x|sin4x|sin6x (910)xtogh2+{TP4SEM4SEES. coh O71)xtogh2~{SEP—SSHEEcoh sin2x.sin6x|sin10x (972)Sar+Ge+tHe 1 xsxt Seol) 00)x39+Saat ==Dee oe x 6m)x-StS -pqteo -S Gn=2warn" 1 1 1 1 08)G-aeb tay aete 076Ea=21431=...co)=94036526 axast, ald O77)loghx+F+5+at © O78)co[logh+2)+GAD4SEEM5oo] (979)loghx+xlogha+2108RO5oo atnase n(n—Vasxe? (89)iogha~Goghaz +~oghay— t°"*n(n—1(n—2)...2-1-ae bd(loghay? VARIOUS INTEGRALS 181 Reference aA 7 =J"cose9.0=f"sine040whereniseven 5.48 50 ==ffoghsinxxwhere0<x<7 X.1420 ==fft0ghcosdxwhere-F<x<3 x.142 lo2 2 ==fFtoghtnxdewhere0<x< x.142 =[cPdewhere2< fo 'T.133Seealso'A.336 “x'T. 134 -fcos(x2)dxwherex2<co o ‘1xe-ldx-[4 7.134 et lel=Sam[i A.336 jt io» -[%~ x -f[s*Bex -[ee~ Fy =[veas 182 SUMMATION OF SERIES Sereno.qxs31x?|1278 Wetsatrsatert Tsote x ex OIF E+E+Et 1 x x OSatitreRt® 2 ® O85loghO+0+3+GEte. noIn-3 2 86)"IAA SG ® 6 #7 08)6-SRtsp te@ 2ow (988)logh=35,+gqi—es (989)Flogh2=2>.(-D S_(-1y02> GreTF 2 wy ate 691) F-2[a +dew+FU+Ie +B+Sac%+...co] iti ix (992)-C=loghIx]+x-5+55]-Gayte? C=Euler’s constant; seeNo. (1132). ++No,(984)hasbeenomittedbecauseitduplicatesapreviousseries—Ed. q VARIOUS INTEGRALS 183 4 Reference xdx-J2a AA.204 ye togh 1ft[d A.294 =r, Toxe %wherep>0 (1907) ; r-f'*foes] A294- J,73 (1907) ¢-fge K.305 2 a [email protected],226 lo0 [yao K.305 =[ya K.305 an=[0ecotoa 22.1875 -f78d) S26alsoNo.(308). 22.246 oind =f°tan(sinhasin6)d8=ts whereaispositive A.517 =f°sdt A.334 184 SUMMATION OF SERIES SeriesNo, \11 1L 1 0)2TigeltxE2~ Heeste eeeee oH>>4...20On+ip 95)>Coatp SeealsoNo.(308). x2 xat on-S5 098)S(-pe'S Bart 9)2>aap 1 1 u- (100) ae tage 1” a05 Ele=nln=2-4 +o ort.) ‘Thenumber ofterms inthebrackets is$nor4(n+1). (00)1-54+5-4Bw Ss nt en (r+Im (1003)«LOG “wsin{0-C4De 111L 008Hisstshytgyto} 1 VARIOUS INTEGRALS 18s Reference =ieetdtwherex>1 260 0 -[ae where|x|<1 AB.165 lox -1=fe where|x]<1 AB.165 oe =ftlogh(1+x)dxwhere|x|<1 AB.166 5 *dx -flogh(I—x)where<x<1 AB.166 -f{logh(1+9)where0<x<1 AB.1665 =fftognp+oewhere0<x<1 AB.1665 xx 1yet-[iewherea>0 A189 “© (sin 8)"={°(22)a LG) A.518 1~fwde 12.135 5 =foresinao 12.135 =f°48286086dd 12.638 186 SUMMATION OF SERIES Saar (4005)12+(i)+(3)+(3)$e. 0008)8[-p35+735+Satstato] (1007Fecm[rice+moe)edge) ++Dol=2)pengteemterms] ‘XXILBetaandGamma Functions. Seealso(1101). 008)faL=xyde fnentde BO, m) Ta+1) na)G)reor(x+Ae(x+2)..a(x+=) (1009)Satpeh (1010)dene -D@=9.-ot. (O11)eeAQmVe{itito ! BETA AND GAMMA FUNCTIONS 187 Reference =Bf”ecocost(sina0—1 2Z,296 -ffsn2009h—2acos@+a?)déwherea?<12Z.308 © cos rx-feye 22.27 =B(l,m) (Beta function) =T(r) (Gamma function) _Porm TU+m) =nI(n) =n! where misapositive integer =Qn)? wae 22.62 =T(nx)2n) o/h 2Z.94 T@= 0 22.59 _Pera)“Teh 0.289 =ae whereaispositive Q.260 =Mx Q.253 188 SUMMATION OF SERIES SereWe poe 4TB 3 (012)+Gwe toexe EDT? 1_a-1, @-1a-2) (013)ye 1t aieFD (@- 1a ~(a -3) SCE) Fo a aa+1) aa+Ia+2) 014)1+$00)+FEEDat4EEOEDatew (1015) IfinNo, (1014), b=1 tears Dey... 0 XXIILInfiniteProducts ay, 2 a 016)at-(1-(1-zm)o ae, aor 402 017)(1-S)\(1-za)(!-4)...es 1018)ae+{cosh$—cosO}{cosh#-con(#+=) «+feosh¢~oos(0¢*=?)} ranch =|]feosh6-cosa+2)) m1 1019) 2dxreosXt3a44 cof (= onto 13 fare +t (1020)(+1)[]{22=2x008245 41 Ut weet} (1021)2-0?sinFin£weesin57. INFINITE PRODUCTS 189 Reference -{[(oe.i" ispositi A. [reey] wherexispostive .524 =FOr@ A.524Te+a) wherexandaarepositive 4 ___Teii=ret .A.294 =Te@re—a J,—t—x1 —%whereb> 4>0 (yoo) =(l- xy =sin® A.213 =cos6 A.214 =coshné—cosnO E.143 =x" +1 where niseven E.143 =x" +1) wherenisodd E.143 =Vn where niseven E144 190 SUMMATION OFSERIES (1023)cos#cos(0+22)...cos{0+m~2} (1024)sinasin(0+22)...sin{o+@—1} (1025)tan7tan2%...tan40—De (1027)26-97sinsinzwegin53* (1028)(1=a(t+5x)(1=Fait+#)se® (eB) AI) | INFINITE PRODUCTS 191 Reference =1where nis odd E.144 =ghrc0sn8wheremisodd -at[((—1)"?—cosn6]whereniseven E.73 =(—1-02 ZAsinndwheremisodd =(1?gh(1—cosnd)wheremiseven E23 =yn where nisodd E.145, =cosnd E.145 =1where nisodd E.145, -_—7 ALIS T+ dor -}) 7 =cos7—sin A.224 =erlotsinax AG.40 sin@ee,=a omitting n=0 A.215 | 192 SUMMATION OF SERIES SotaOecoos)T{(:-Zem(1 +Scum} (1033)intfencrrn-am} (1034)TH ns(1035)Tl(oyvost 4037){(:+yi+(=3)} {i+ES) +(eS)}H +(gig)}--% (1038)cos$-0s£008$5...90 (ao4oy(1~S52)(1-Ss)...(1 -aean (1041)21Ffsin(0+ra) (10)Lstnfan. INFINITE PRODUCTS 193 Reference sind = Q.137 =cos6 A216 =2 3 =x! AD. 10 -[en ap.18 nme _coshk—cos@ =T3088 Q137 sind -% Ald =elle 0h2sin8 Q.35 =cosn@ (neven) bas A.2 =sinn® where0<6<a A211 =1 12.355 | 194 SUMMATION OFSERIES Series No. aostef(r+S)(r+2)%(0+3) 7" (1044)zee](1+a)one (1045)2-1TTsin(0+2) ro (4046)2sin(0+F)sin(2+2)...sin(0+=") (1047)2-1088cos(8+2)cos(0+2)ces(°+aot*) (1048)21cosZ00s3cos3...cos75 (1049)2-1sinZsin3%.sinte (3050)0s“cosss608@ (1053)I{i+ura (1054)I{i-ta} ; INFINITE PRODUCTS 195 Reference =e 1.355 _sinn(x +6) A204sin we 7 =sinnd E145 =cosnd E145 =(—1?sinndwheremiseven =(=1)9? cosnd where misodd E.145 =cosF E.146 -1 E,146 (==-e E146 _sin(a-8) “Sina E.159 sin(a+8) ‘ ‘ vei=2G+9henrisapositiveornegativeintegerorzerosing E.159 =ses whenrisanoddinteger,positiveornegative E.159 =512—9henrisanoddinteger,postiveornegative E159 oss{1-(Fa)MtYH (wa)} (1056)(1-Sy-(=e)Ht-(=)}--- © (1059)(0-#)(!-all-a© (4060)ina (4061)ITee (1062)eos${1+(B)} +(4)}- ~ (1063)(1+at+x!+Reo (1065)(1+i(t-5)(!+3)(!-Aa~o INFINITE PRODUCTS 197 Reference £088+cos@ ~TFesa E159 =sesdaosa F159 1=cosa =sing=sind E160 sina _rant=Tee? Q.259 =F A.106 _Td+900 +wl +Px), ~T= xr =a) =Px) whenf=Bl+iv3) orp=1 A313, _Td +ar +4) “Ti+a+6) Aults =2cosh@ +2cosa E.160 =}sinh» E.160 a__sinfeVa? 3)Veee sinza E16 =I1 A.108 8 Ag vg =-R72oFLFvals feeltel! ;1alsoffsftgleg ¢=‘ESFEE ZogSS Re OG3 a8 eo ite ££! ! Zz =: SRA ATAzZok ae +Boag >$i: 1h8 SiMhieGeaE>sSa mS ES pire}gpRRaSPe sak: 7weeSeeSPS he eGR|2 SSTeaREfcBalahahE* ff!ifor ft& oP rae‘aaa =writ 7 :BOS RS ~See ty aif §3F3=a 5a=3aeleif2 222s € EF €&E Bs 2228 2 8 = | INFINITE PRODUCTS 19 Reference = A.106 =0 C159 =} A.106 -}. =0 E155 -3 A213 eete E.190 =0 Ail =SMEwheremisodd A.210 =Sind where miseven A210Hind cos =2058 wheremisodd A211 088 =c0sin@ wheremiseven A.2iL =£08n8~cosnf 1.84 200 SUMMATION OFSERIES (1077)I{er~2abcos(0+2)+v2} =a (1078)TTC+a) * com)|](+ara} = 6 (1080)T]omx: (1081)[Ta+x2)0 oomTT[-@Fap] (1083)in[i-wal (1084)Tlfeom(1-Fa) XXIV. Fourier’s Series oss)£22)FO)—fon= KO (1086)5(0)+4(2)+4(2)tet$00) aos 50)+4(22)+...+3/00} | FOURIER’S SERIES 201 Reference =ain—2arbrcosnd+be 7.84 =sinh@ T.130 =cosh@ T.130 =ne T.130 =oywhere? <1 1.130 =1-Stee A.224sin? we =—ndx A.224 sinx =88n=omitted A215 te=f‘f(0)cosntdtifalldifferential coefficients arefiniteand0 theseries isconvergent X.140 =zi,"dt+2s10,c0smede a0+E[Acosamd+...c0 X.140alo _pe moot it-ffl)dtwhere=1+a+ate© andmLtetpte.pete. x.140 202 SUMMATION OFSERIES Series No, 1 1 Ban —oy+Ajram -FO) =ZrUam—FO}+PATh—PO} sn-EEVm-£0)+...© Series tobeconvergent (1089)A570)+foe+)+Fox+20)+...)+AS09 -Agro +Are -Bae +... Seriestobeconvergent XXV. Hypergeometric Functions (1090)1+fox+oeSa ae aa+1a+Dob+Ib+2) +Bide+Ife#2) Tele -a- 6) (0)Te arte =8) (1092) F(A,1;C,x) xml m+2.. (m+2)...dm+ 2-2) 1083)eTmea tre 3).+2Be} 1(m_+2)(m+4)...(m +2n) 402 GS im+3). +2a=1) af1x[moaPA)—5d HYPERGEOMETRIC FUNCTIONS 203 Reference =[roa x.140 =[row x.141 =F(a,bs6.x)where |x|<| Q.281 =F(a,b;¢,1) Q.282 1 ck=pAe- 44) Q.286 =(1-xnfim(l=#2)" wherexand1—xarenot 0 negative realnumbers Q109 204 SUMMATION OFSERIES Series No. XXXVI. Relations between Products and Series (1094) (1+x2)(1 +x32)...(1 +x2emtz) (1095) logh (1—x\(1 —x2\(1 —x3)... oo} 1 2)(1096)II(1+2) (1097) [](1+x2) 1 (1098) T](1+x2) (1099) Inthefollowing relations between series and products, all pare prime and may berelated tothePy,dqfmand fy ofseries Nos. (315) to(318). . 1 1 L 1Series(315)1+5-7,—TintTyeele a 1ui11Series(316)1-5—35+TetTg=hn a L 1 1 1Series(17)1455-5 —Getapes=Pe a L 1 1 1Series (318)1Reet ROT =I | PRODUCTS AND SERIES 205 Reference SoCL=x2m)(1=x22),(L—x2m-2h)eran wie2 = CMa =--> Seay=where|x|<Iandf()denotes thesumofall the divisors ofthe positive integer n;for example, SO=14+244 C.345 ea SEAPW +Pd) +Ped ipthe =1l+2 Dbbebe iftheproduct converges toadefinite limit C.420 ng SLE Nd =1). =OY) ns<4 Sey en Cc.340 < xlnlet)/2g0 =1*aa where|x|<1C.341 AE. 87 _1 “ I 1 1 1(1-5) +a(t+re) ae) 1 = T T T T(+s)(t+a)(0~a)(!~ a) 1 (ia(t+B)(1+B)(1=ppp)+omittingnon-primes 1 (1+Ft+y(t-NG+in)++»omittingnon-primes | 206 SUMMATION OFSERIES SeriesNo. (1100)T](1+»,) XXVIL.SpecialFunctions r@_ 4 (1101)x)=Te)~aoahT(x) C=Euler's constant; seeNo. (1132). 1 1 1 We)=~3-4 eB-a+xen+1D d(-4-)aNeen eT S(t 12Ga-=) LeS¢--4) sorAx¥ i Su y loghn+>qu+A) wetd=tewe Wl—x)=Wx) +meotmx 0)=—0 | SPECIAL FUNCTIONS 207 Reference < om 1 ~t4Dmwhereit=ape Pe T.131 and theseries isconvergent pee ; iti ==C+ [FRatwherexisarealandpositiveinteger 5 gueT.132 Ix-1, 1 _x-1wnceptety Lee 22.9 =-c- 4) 7.133 =Ha)—Hy) A.522 1 2Z.100 HVQ)+C+> {Fin =Hy(z+4) _(2),==Hue) +(3)}=Ax) 7.133 tp-ficota A523 =u) 22.98 +3)==C—2logh2 (3)=2-C-2togh2 A522 5) _845)=§-©~2togn2 : 208 SUMMATION OFSERIESSeriesNo.W)=-C ¥Q)=1-C ¥@)=3-¢ 1vt Wa)=(14549)-¢ ta, ot w=(l4z+ged-c worePiyaere: per+)BQ)= BCs)+BUx)= All)=logh2 BQ) =1—logh2 (3)=~5+logh2 11BG)=5+5—logh2 1 7 i 1(3)-3 (3)=tosh2+32v3 5 2 7(3)3-245 16%toghPe)=ve) | SPECIAL FUNCTIONS 209 Reference 2Z. 136 i 343)==C~Flogh 3 72y3 3 WQ) ==c=Sivan tH ® 22.136 v(3)=}-C=31082 ra 43)=-F-C -3log2 T.133 A. 523 St=Lare 2Z.98 210 SUMMATION OF SERIES SeriesNo,¥® Wa htet hte YOnptet pte i ¥6)-4( +A4) VO=ht pt gt. | ¥(~0) i 1102)Wageye=(3)-«o DlopSal?2") SlvvdeeNl"2) (28) 1 SPECIAL FUNCTIONS 2 Reference =o 2Z.100 2 “3 2 “% am -F-4 2 =5-1 fm? 11)_mt <4(F- pox) 5-44 Ps “67a =0 =aed 4D T.19 at(T=)wal >’)? r—p—l)\,,.-ret (3) =m T.19 =0 T.19 '2n)=(7) T.19 | 212 SUMMATION OF SERIES SeriesNo,(CE) +CE)+f) 1=(i)+(3)-.oe(2) Seealso No. (189), etc. TableofBinomial Coefficients (j=nan (i)(3)(3)3)(8)()@)G)(i) (2) 21 3°93 1 4641 51010 5 1 615 20 15 6 1 72 35 35 2107 «20 828 56 70 56 28 8 1 9368412612684369110 45 120 210 252 210 120 45 10 1 11 55 165 330 462 462 330 165 55 11 1 12 66220 495 792 924 792 495 220 66 12 1 XXVIII. ZetaFunctions (1109)>aw? Ifsisanegative integer =—m (-2m) {1—2m) 0) | ZETA FUNCTIONS 213 Reference _(ntl 1 ~(i+i) ne -!="; ) T.19 T.20 =Usa) 1 pexetee=| —— ).266Tahize* Q =0 =Cirse wherem=1,2,3,ete, Q.268 --! “73 » | 214 SUMMATION OF SERIES Series No. XXIX. Legendre Polynomials (1104) IfPo(x) +APA(x) +HPPAX) +... WhereP(x)=1,AQ)=x,Pie)=GP— Psa)=(5x?—32),Pala)=$G5x4—308?+3) Pax)=}(63x5—70x?+152) and (2n)! nn=1) Pc)=BOG[ew-HERD art nla—In~Xn—3) +ee. ] then S (ayyn=20! ao aria—Nn=a Also PAI) =1P(—1) =(=1h Pagts(0) =0 | op BeeQn=0) Prl0)=(—1)"3G “Gay Finally, P,(x) =wisasolution of Px)=F(n+hamtsb=ba)=Pop1(X) XXX. Special Products (1105)Ifgo=Tra—q*), a=[[a+a% 4 SPECIAL PRODUCTS 215 Reference =(1=2xh+W212 where[2xh—|<1 Q.302 =Ps(s)_wherem=2or=whichever isaninteger Pew da a993-2 #+nlnsNw=0 Q.304 -pycesDn+2)...+A=Ml=n)...-1m) oS aa x(i-ey) Q.312 Thengogs=[[(—9")where|g|<1 A1I6 1 aaa=] +9°)where n=1,2,3,etc. 216 SUMMATION OF SERIES Series No. a=[[a+er) a=[[a-¢r) , The four products qo,di»42,and qsareabsolutely con- vergent. Forexample, Qa aqaye (=gl =gt =49)... oI?=1+2g+2g+29?+...2 Gon?=LtGtgh+get... gn+...0 ge1=292+2gt—2g!+...+2g+... 0 7 qos=1-9+g)+GS+9)—GIP+")+... a8=938+16qq)* qo’=1—3q?+Sq6—7g"?+9g... XXXI. General Forms S onan (109)2rae+wD Ifthedegree of¢(n) isless than that ofthedenominator, resolve into partial fractions and use <cn 24- wwe{EeSet +tosh-»} i Seealso No. (370). GENERAL FORMS 217 Reference 19293 =1 =(14gil +gl +99)... © dots?=1=2g+2g=24?+...co gy?=1=34?+Sg6—1g}?+992—..,co Walageetartart...tginimmine,.. o3 indicesbeingalternately oftheform5n(3n+1) This canalways besummed ifconvergent. c.246 a,b,...karepositive ofnegative, unequal integers and ¢(n) is anintegral function ofn. Ifdegree of¢(n) isgreater than that ofthedenominator, itmay bewritten x(nxt HO"+Gas DnB where thedegree ofx(n) isles than that ofthedenominator and =n)xn may befound bypartial fractions and(370) | 218 SUMMATION OF SERIES Series No. S $n) (ton)2@Far+5)...+B) IfinNo. (1106) x=1,theseries isnotconvergent unless thedegree of4(n) isless than thedegree of (n+ @)...n +k). S yet Ha) (1108)2 ‘GFaa+bs Ifabsolutely convergent, Sum $; Ifsemi-convergent, Sum $2 Intheabove two, theseries isabsolutely convergent if thedegree of$(n) islessthan (n+a)...(n +k)bytwo units, and semi-convergent ifitisless than (n+a)... (n+B) byone unit. (1109)>4,05 canbesummed2 1 The identity can beestablished $An) >dela Then>eae i GENERAL FORMS 219 Reference rolit L “5 wafi+aty+- 3 m8nO C> Ea) where >meansthesummation withrespect to 4a,b,¢...k, anda,b... karepositiveandunequalintegers, and¢(n) isanintegral function ofn @,b... kareallpositive integers and¢(m) isanintegral function ofm. C.253 11 al -,2og ate cpet " @— aye ay... >a = (=1)"e(—a)=$1+logh?20-H —a)...R=8) if(n)isanintegralfunctionofmofthe rthdegree C24 (and seePart I,p.107) =Ay+Ayn+Agr=1)+.Agni=I).=7#1) x0 yet Set=wri+40Gyat482Gat a+Axe2a =(Ay +Ax +AR? +... Aer 20 SUMMATION OF SERIES Serles No. Todetermine theconstants Ao,etc., divide 4,(n) byn $n) divide 4,-1(n) byn—1 $10) $0) (1110)>)4.40)Cu" D ‘The identity canbeestablished asin(1109) #40) ‘Then thegeneral term intheseries is GAM) CnX"+AgmCX"+AyCyX*Hee +Ayn—1)...=1+VD)mCnx" Therefore Ddale)mCnx"=Ao2muxt 5 +MAX>mCXEHoe +m(m—1)...(m—r+ ALS Cera ‘The constants Ao,etc., may beevaluated asinNo.(1109). S40)aC OUDGyo +b)...@+® This can ingeneral bereduced to SHin)winCmin2°(m+1m+2)...(m+kt GENERAL FORMS 221 Reference =$-ilr)n +Ao =b-alnin —1)+Ay =An =1)+Art where ¢,(n) isanyintegral function ofmoftherthdegree C.195 =Ag+Ayn+Agn(n—1)+... Ann=1)...27+1) Ag CyX*MASKmCXE Fmlm—WAZ?py2Cy22?+o malo —Mo.Qt=1WAGae}rXO =Ao(l +x)" +mAyx(L +x) +. +mm =1)...(m =r +DAL +xr . mAyx mm —W)Aax? - -r+ Ae} +m(m—1)...—1DEES rp+ where a,b...k areunequal positive integers inascending order ofmagnitude. where y(n) isanintegral function ofnviz.4,(n)multipliedbyall thefactors which arenotabsorbed bymisCut- C196 22 SUMMATION OF SERIES Series No. (112) ¥arg(x) The series within thebrackets stops atthenthdifference of(x), supposing ¢(x) tobeofthenthdegree. (x) isrational and integral (1113) Sum ofaseries whose nth term isr#times anintegralfunctionofnoftheSthdegree,suchas Ddlar’+ayn!+...a,}re Multiply by(I—7). For example, (1=>(2r+22+...mre) whence byaddition Der +Br+...nr) (1114) >uy(Approximate summation) The constant Kistobedetermined ineach case by substituting aknown value ofx. (1115) Ifthesum f(r) ofafiniteorinfiniteseries F() =a9+ayr+azr* +... isknown, then 4y.c0sx+ayrcos(x+y)+azr?cos(x+2y)... aqsinx+ayrsin(x+y)+ayr?sin(x+23)... +For valuesofBy*,seeNo.(1129), 4 GENERAL FORMS 223 eerence =0+{4a -4He Ww.53 @ 2 4s +ep -Gp aH---} where 4¢(x) =d(x+h)—4(x) =Prt 27? + +... ee -3P -3-2P... +3-PP +... =Lr...=nets TH =(+Wrett+(2n?+In—Dp=eed a= =KtSuede—Sue ‘By*du,By*du,|Bs*d5uy+e ee Be. 22.117 =Sletstrety +evptre-y) =—Hepirer) —eye» T.81 224 SUMMATION OFSERIES Series No. (1116) Euler’s Summation Formula De) =(0)+£2)+...+ 0) When f(x) isarational algebraic fraction oratrans- cendental function, this cannot necessarily beused, and theright-hand side becomes aninfinite series which may not converge. (See Bromwich, Chap. XII, for anumber ofapplications ofthis summation.) (1117) If1,1. Lf, etc, arethemagnitudes ofthedifferences ofthediscontinuities ofthefunctionanditsvariousdifferentialcoefficients atpoint a,and similarly forpoints 6,then a ly), ly,~qdlasinna —55>hy!cosna+=>>1,"sinna Ly).+ablecosna—...co LYheosna-ASWsinnaBLMcosna ly).+polesinna+...© XXXII. Double and Treble Series «aiis)sSoan 2,2.mint \2, SEStns! “rr (nis)>>>are & ' DOUBLE AND TREBLE SERIES 225 Reference =Sf)de+3/0)+HBS)—HBS) +A308 where f(x) isapolynomial andthere isnoterm ontheright-hand side (initsfinal form) which isnotdivisible byx. =7a, (Fourier’s nthharmonic amplitudes) =xb, x.57 20wherea,=4"Aopcosmt dtandb=1{7(sinmeatalo alo andf(t)=B+ayc0s+a3.c0s21+a3C0831+... 4bysin+bysin2r+bysin30+... =k whee-2<x<1 A194 =_y whee-3<x<1 A.194 226 SUMMATION OF SERIES Series No. SF (1120)22wr’ ae 4 a2)>>wR Se 1 (1122) wae22Gs—2 Sey(1123)2>oud >< 1 (1124) @-22@s=1 Lim< 1 norS[3as] ALA tes 1 zy>2.GeaT XXXII. Bernoulli’s Functions (1128) Bernoulli Functions. Valuesof¢,(x) $(X) SeealsoNo.(1129), +See No.(1128) forvalues of$x), BERNOULLI'S FUNCTIONS 21 Reference - A194 pe =~}hogh2 AL194 372 =f A.194 =logh2 A.194 Togh 2 A194 =Jlogh . ==n? allvalues m=mareexcluded A225 3 vee ,=2?12(by)—$20)}+g(y—5)] omitm =n A391 4 aly 3 =0ifx=ywheed<y<x<} bad thxb Q136 =Fcothacotha . MetgMORDpgmln=Din=2X0=3)p arFat OD pyrEO WD) pays A.300 terminating either inxoFx2, 28 SUMMATION OF SERIES Series No. Therefore $x) =x $() =P =x 35,1 wav a4!AG)=5+5 $4(x) =x4=28 $x? S44 5,3] $s(2)=8—5xt+3x—2x 541 : dela)=x6—3x84 Sx—5at $x +1)—G(X) =nixed dx += ex ete. This function isrelated tothat inNo. (1135) by BAX) =$4(3). (1129)Bernoulli's numberscanbecalculated fromtheexpression 2m! fl 1dapistmete to} Asimilar confusion arises inthecase ofEuler’s numbers, No.(1131). Some authorities, forthesame value of1, quote thevalues ofB,,and others Bz,1. Inthetables following and inthisbook B,values areused. For example, inAC. 42,A.297, C.231 and AE. 3,ete. 1 1 1 Bab Bay B=gyete whereas inY.503, ete. 1 1 ped BY=i BY=syBet=fyetc. where B,=Ba,-1*. BERNOULLI’S FUNCTIONS 229 feference 4,(2x)isthecoefficient off,/n!intheexpansion of¢aotA.300 C.231 C.363 By=Bay-s* ns\y 22m)!(+) \ hi Im)On, 4yA \2—1)1 BO 5taf | 230, SUMMATION OFSERIES Extract from British Association Report, 1877 (Adams) Table ofBernoulli's Numbers expressed inVulgar Fractions ‘Numerator Bo=-1 Denominator No. 1 61 1 302 1 2 3 1 3004 5 6S 691 27306 7 67 36175108 438677989 174611 33010 854513138 2363 64091273012 85 53103 6 13 23749461029=87014 861 58412 76005 1432215 770 93210 41217 510.16 257 76878 58367 67 26315 27155 30534 77373 191919018 292999 39138 41559 6 261082 71849 64491 22051 13530 20 15 20097 64391 80708 02691 180621 278 33269 57930 10242 35023 69022 5964 SII11 59391 21632 77961 282-23 560 94033 68997 81768 62491 27547 46410 24 49 50572 05241 07964 82124 77525 66 25 BERNOULLI'S NUMBERS 231 232 SUMMATION OF SERIES Table ofBernoulli's Numbers expressed inIntegers andRepeating Decimals No ta ow v2 —00s 63 409 a — oo o5 oats | €6 —02531135 (7 118 lee —nonisc4ens008[es seancss 2 9.14 22" ga.ra18e4s79710144927598 bn —sss02851138 | ho a2ssi218 230 —rragenn0671cos5407298850574712643 3615 401590473,96064236884303681748359167714 (bu1511657670157anno to 42961464006118, i \Srear ssseosonn.38771590345618| ta ‘ensue net | VO20—192965793419 anak25684“(ua {416990757362615 00055370980354374307862 679955703211517165 !LL wonmortasanseassers.res 1s9e2css4ssor ass ; GG25. 211soranesoso1 sea.14s3ore219881560283687945262413475177049 :UZ21208626s220965259346027.3193 70625253178194354668942900257017884076707608: (251908rus07963685720.018 Inthe“Report oftheBritish Association fortheAdvancement ofScience, 1877,” page 10,etc., Bernoulli’s numbers arecalcu- latedupto=62.Further,themethodbywhichtheyare |calculated isalso described. ’ BERNOULLI'S NUMBERS 233 234 SUMMATION OFSERIES Series No. ‘ValuesofConstants usedinSeriesNos.(305)to(318) (1130) EPP To B,\-1|6|1/30) 142) 1/30 5/66«| o |r [a 3 7 155 Bj1|u3}75}31/21) 127/15} 2555/33m}o |r|213 164 33558|2|1|13s|12/7|1093/5 49205/11«|-1|1|915|3751/7|1388t1/s| 25143755/11&|0|1[13|363|18581 1525355m|0|114|403|20828 17144056|0|1|10|273|13940 1144055E,rf1ft25 16 Es|ue|i|se|oie|1385* 50521*R,|1|7[30s[33367 f68.15585 |2237423527S,|1|520522265 4544185 |14916325251|2)1B]47 809/9 1847H,|32|3[33|903|46113 3784503Jn|2|10/3/34 |910|415826/9| 3786350T,|0|1|23|1681|257543 |67637281P|1|3157|2763|250737 |365815232,|01ju361 24611 2873041 Forthetheory ofBernoulli polynomials, etc. including very many practically useful recurrence formulae, sce Glaisher 1898a, c,and Lehmer 1935, 1936. The numbers By,ImTmPy»and Q,are fundamental and arenotexpressible simply interms oftheothers. It istobenoted that the odd Eulerian numbers are not calculable directly and are fundamental, See Edwards Differential Calculus, p.502. SeeNo.(1131)forexplanation ofE,andE,*. / : VARIOUS NUMBERS 235 Reference AC. 42 236 SUMMATION OF SERIES Ba 5 * be 5, Ay In A, tn In Py 2, See No. (330) forq,. +Foracommecon ewanBn!)estNo0), VARIOUS NUMBERS 237 Reference =22 —1)B, AC.41,42,and48 =2" —2B, 3 -ried —1B, =(3% —3)B, =}O™—20-92, =;(22—2)(3%—1B, =Fe—NG—9B, =f —NG—DB, = +Dh = +Dh, =H, +I, =etetBaa(i)t AE.SI -0e{o Ban)-3Bas(3)} AE.51 =(-De Bs(5) AE.SI =(DeeBina(§) AE.SI =DME,* —(2): 2B,4*.(=I 12n)2E,* +(—1PEo® -Puan (@)-es] AE.62 -Sa-w()" AE.64 238 SUMMATION OF SERIES Serés No. R S, qT, (1131) Euler's Numbers arecalculated from theequation gered a 22m)(2){ua-artgerneo} aye 1 1 1=200m)1(2)"" (1+zm)(! -#)(!+pm) Inthelistbelow ofEuler's numbers odd values ofE, have been included forcompleteness. These have some- times been called “Prepared Bernoulli Numbers,” and have been omitted from many lists, and only the even numbers included. ‘This has led toconfusion because insome cases the even numbers were then called “Euler's Numbers E,” In2Z. 243 and Y.$01 Eulers’ numbers are asshown in the long list below; but inAE., AC. 42, T.141, and C.342 and 365, the numbers are shown asE,* where Eo =, ne0123 4 5 Eg =1 1561 1385 50521 ete. andthroughout thiscollection E,andE,®have been used todistinguish between them. Eo=1 B=1 E=2 &=5 Es=16 Eg= 61Ey=272 Ey= 1385 Ey=7936 Eyo ~50821 (1132) Euler's Constant C=0.57721 56649 01532 86060. .. EULER'S NUMBERS 239 Reference =hom+nes AC.50 =} +DEF AC.50 3S, =2R, +Ex* =2"an—118)" =3125-0n- w1() AE.75 ©.342 .365 Y.502 =Ef =Eq . Ej=270.2765 Eyq=199360981Eyg=19391512145 Ey=24048796 75441 Eyy=37037 11882 37525 2.245 22.87 240 SUMMATION OFSERIES Series No. 1 (1133) Sum ofPower Series sapthetene Values of$,tosixteen places ofdecimals aregiven inthe table. n §,tosixteen places ofdecimals 1|0.577215664901532B60EStog co(Euler'sConst.+00)2|1.64493406684822643|1.2020569031595943 4|1.0823232337111382 i5|1.03692775514337006|1.01734306198444917|1.0083492773819227 i 8|1.0040773561979443 i9|1.0020083928260822 j 10|1.0009945751278180 11|1.0004941886041194 12|1.0002460865533080 __Thesixteenthdecimalplaceis 13|1.0001227133475785notalwaysthesixteenthoccur- 14|1.0000612481350587ring,butthenearestinconsider- 15|1.0000305882363070 ationoftermstofollow,e.g.C16|1.0000152822594086hasforits16th,17th,etc....17|10000076371976379 figures8606.... |18|1.0000038172932650 Log,10=2.3025850929 20|Leonosoens9 manent 0|1 ° =C=21|1.0000004769329868PulersCont=Sioss.acoe.22|1.0000002384505027 — 23|1.0000001192199260*=04342944819... 24|1.0000000596081891 25|1.0000000298035035 26|1.0000000149015548 27|1.0000000074507118 | SUM OF POWER SERIES 241 Reference 22.144 242 SUMMATION OF SERIES Series No. ; n $,,tosixteen places ofdecimals 28|1.000000003725334029|1.000000001862659730|1.000000000931327431|1.00000 0000465662932|1.00000 0000232831233|1.0000000001164155 34|1.0000000000582077 | 35|1.0000000000291038 | Relations between Bernoulli's Numbers | (1134)Pt+21+ett Geet 1 amo -1i (1135)2=Sat+ABet-@=Dera=3)Byte. 1 The last term is i Thelasttermis, i (1136) Baril —») \ Ba(l —x) | 2,0)BQ) i 1 (1137) Bx) —(n—DxBy-y() +(rn—1)2x?By-2(2) ! fee. (=1y 22x)+(1b | ~po- - |(m,=a=XeDaanr+1) (1138) 12+22"4324... Qe) 12+32+...(2x—1" Dmdnt..+(2xp 1 | BERNOULLI’S NUMBERS 243 Reference =B(x) where xisapositive integer AE.8 A. 304 =B,Ax) patsByn-1x?_whereniseven (DB y-pyaxwherenisodd AE.7 =—Bais(2) AE.4 =Br(x) =0 =0 =—-B(- x AE. IL -.c-() G =Brix +1) =Bagsy(2x +1)—2"Bopey(x +1) AE. 15, =PBanr(x +1) | 244 SUMMATION OFSERIES Series No. (1139) 14,00) Ar(X) Ad =») Aryl) A2,(0) (1140)&isapositive integer 4s)+Bef) +.B =) cana,(Z)+8,(Z)+---a{&) E (ai42yBade) of) +f) nf3) sul) Ae) Hi ' BERNOULLI'S FUNCTIONS 245 Reference =Fpm+(Dabur? —(Bx, ete,theseriesbeingcontinued solongastheexponents arenotnegative AE.19 =Bass)+(-1yrBe AE.18and20 =(=A, =Are(l)=0 =Aft)=(9Bt =peiBulkx)_ wherenisodd AE.9 =phamtesy+p ELIE amine =0.wheremisodd AE.9 =(1pBatFewheremiseven 1 1 )\B,=Cir{l+gtasah AE.43 1 1B, -Cr+g-galt AE.31 3-13, =oBate AE.36 =(-yt AE.26 1 1 1 1 1=5(1+3)Bov(§) +parsiBaen(Q) AE.74 1 i-2Boru(}) AE.63 | 6Bye(2) ~3*Bae) 2a(3) : Boou(3) Baeu(3) ali)+(3) Aula)+449) ; “ofs) Andi) 43) 4nd) Qn+DHBaer(§) | BERNOULLI'S NUMBERS 247 Reference =(-IeHH, AE. 51 =Azen(3)=egs AE.31 =Aawn(3) =GO AE.35 =Aann(3) =0 AE.26 ==Bawn(9) AE.63 =wie) AE.77 =weiAa) AE.66 ~orbs-gs} AE.31 =(r{i~ sis} AE.36 =(7th AE.26 ==14eneD(1-3)a AE.57 =@n+Deat{i=5)B teGDMOn +D1=55)B, 248 SUMMATION OFSERIES Series No, +H, 1 Qn+16%Bari(3) ‘Thederivation ofsomeofthesenumbersisfromsuch seriesasNo.(576)andNo.(577)byputting@=?;r etc.;buttheoriginal articleinAE.should beconsulted for afulldescription ofthederivation. 2 nt t (1143)3{to~HasBat-... o} (1144)3.1,~Qala +Qrduler ++++(DoQmah,+(Ihe | (1145) Et—Qn)aEga+QmeByat +... +(DM Ome +(=DE InNo.(1142), seeNo.(1130)forvaluesofJ,E*,andH. n,isthe binomial coefficient nn —1)..(n— 1+1) rt (1146)28,5)=-5 i 1 i 1 ep(!) —51m3) =34 | BERNOULLI'S FUNCTIONS 249 Reference 1==1+n+(1=5)2 AE.51 1=Qn+De(1-3)By tet(EDOQn +y6e(1-=)B, =|_ AE.37 “Tretee 7 =0 AE. 37 =0 AE. 37 v4x(3)=-é AE.26 i 7243)= i 31243)=~ins i 127(3)= | 250 SUMMATION OFSERIES Series No 32,(5)=1 3*2,(5)=4t ea()--4 sai)=-3 | BERNOULLI'S FUNCTIONS 251 Reference v4,(3)=-4 AE.36 ai)=-3 AE.3I AE. 61 AE. 61 | ' ACATALOGUE OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST oo ACATALOGUE OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST War IsSerence?, N.Campbell ‘The role ofexperiment and measurement, thefunction ofmathematics, the nature ofscientific laws, the difference hetween laws and theories, the limita- tions ofscience, and many similarly provocative topics are treated clearly and without technicalities byaneminent scientist. “Still anexcellent introduction toscientific philosophy,” H.Margenau inPhysics Today. “Afirst-rate primer‘deservesawideaudience,” ScientificAmerican. 192pp.53%X8.(60043-2 Paperbound $1.25, THe Nature oF Licht AX GoLour iN rHe Ores Atm, M. Minnaert Why areshadows sometimes blue, sometimes green, orother colors depending fonthe light and surroundings? What causes mirages? Why domultiple suns land moons appear inthe sky? Professor Minnaert explains these unusual Phenomena and hundreds ofothers insimple, eas)-to-understand terms based ‘onoptical laws and theproperties oflight and color. Nomathematics is required but artists, scientists, students, and everyone fascinated bythese “tricks” ofnature will find thousands ofuseful and amazing pieces ofinforma: tion. Hundreds ofobservational experiments aresuggested which require no special equipment. 200illustrations; 42photos. xvi+36app. 5% X8. 0196-1 Paperbound $2.00 “THeStRaNerSroRYOFTHEQUANTUM, ANACCOUNT FoRTHEGENERALReaver oF THe Grown oF Toras UNDERLYING OUR PRFseNT ATOMIC Kyowtrper, B.Hoffmann Presents Iucidly’and expertly, with barest amount ofmathematics. theprob- lems and theories which ledtomodern quantum physics. Dr.Hoffmann begins ‘with theclosing years ofthe 1gth century, when certain trifting discrepancies were noticed, andwith illuminating analogies andexamples takes you through thebrilliant concepts ofPlanck, Einstein, Pauli, Broglie, Bohr, Schroedinger, Heisenberg, Dirac, Sommerfeld, Feynman, etc. This edition includes anew, long postscript carrying thestory through’ 1958, “Of the books attempting an account ofthehistory and contents ofour modern atomic physics which have ‘come tomyattention, thisisthebest.” H.Mangenau, Vale University, in ‘AmericanJournalofPhysics. $2tablesandlineillustrations, Index.275pP- BMRR 20518-5 Paperbound $2.00 Garat Loras oF Movers Matuestatics: Titre Nature ano Ust, Jogiit Singh Reader with only high school math will understand main mathematical ieas ‘ofmoxlern physics, astronomy, genetics, psychology, evolution, ete. hetter than many who usethem astools, butcomprehend little oftheir basic structure. Author uses his wide knowledge ofnon-mathematical fields inbrilliant ex Position ofdifferential equations, matrices, group theory, logic, statistics, problems ofmathematical foundations, imaginary numbers, vectors, ete ‘Original publication. 2appendixes. 2indexes. 65ills.g2app. 5%x8 20587-8 Paperhound $2.25, | CATALOGUE OF DOVER BOOKS “Tw Music oF rite Spmenss: Te MatsaiaL Univense-—FKom ATOM To.Quasar, Sinriy EXPLAINED, Guy Murchie vast compenditm offact, modern concept and theory, observed and calculated data, historical background guides intelligent layman hrough thematerial tiniverse. Brilliant exposition ofearth's construction, explanations formoon's raters, atmospheric components ofVenus and Mars (with data from recent Ay-by's), sum spots, sequences ofstar birth and death, neighboring galaxies, contributionsofGalileo,TychoBrahe,Kepler,ete;and(Vol.3)construction oftheatom (describing newly discovered sigma and xisubatomic particles) theories ofsound, color and light, space and time, including retatvity theory,{quantumtheory,vavetheory,probability theory,workofNewton,Maxwell,Faraday, Einstein, deBroglie, ete. “Best presentation yetoffered tothein- telligent general reader” Saturday Review. Revised. (1967). Index. $19 ills trations bytheauthor. Total of3x+644PD. 534X814 21809-0, 21810-4 Pwo volume st,paperbound $5.00 Fours Lrcrunss ox Retativiry avo Sracx, Charles Proteur Steinmetz Lecture series, given bygreat mathematician and electrical enginee, generallyConsideredoneofthebestpopularevelexpositions ofspecialand.generalFelatvity theories and related questions. Steinmetz translates complex mathe tnatical reasoning into language accesible tolaymen through analogy example And comparison. Among topics covered arerelativity ofmotion, loation, time; fofmaar; acceleration; dimensional time-space; goometry ofthegravitational field: curvature and bending ofspace: non-Eutlidean geometry. Index. 40 illustrations. x} 142pp. 536% 8}4- ‘ITIL Paperbound $1.35, How 10 Kxow str Wito FLowsxs, Mrs, William Starr Dana Cassie nature book that hat introduced thousands towonders ofAmerican wild lowers, Color-seaton principle oforganization iscary touse, even by those with nobotanical taining, and thegenial, refreshing, discussions of history, folklore, use ofover 1000 native and cacape flowers, foliage plants areinformative aswell asfntread. Over 170fll-page plates, collected from ‘Several editions, may becolor intomake permanent records ofind, Revised toconform with 1950 edition ofGrays Manual ofBotany. xii <p438pp. Sux By 20882.8 Paperbound $230 Manat oFTHE Tarss oFNoxti Autraica, Charles Sprague Sargent still unsurpassed a1most comprehensive, reliable study ofNorth. American ttee characteristic, predie locations and distribution. Bydean ofAmerican dlendrologits. Every tree native toU.S, Canada, Alaska; 185genera, 717species, deserbed indetait-Ieaves, lowers, fruit, winterbuds, bark, wood, growthhabit,et.plusdiscussionofvarietiesandlocalvariant,immaturity variationsOver 100 keys, including unusual ti-page analytical key togener, aid in identification, 783 clear illustrations offlowers, frit, leaves. An unmatchedpermanentreferenceworkfoallnaturelovers.Secondenlarged(1936)editionSynopsis offamilies. Analytical key togenera, Glossary oftechnical terme Tex. 285 illustrations. 1map. Total ofRapp. 534 x® 20277-1,20278-X Two volume st,paperbound $6.00 CATALOGUE OF DOVER BOOKS ‘Tunex Science Fiction Novets, John Taine “Acknowledged bymany asthebest SFwriter ofthe 19208, Taine (ander the name Erie Temple Bel) wat also aProfesor ofMathematics ofconsiderable renown. Reprinted here areThe Time Stream. generally considered Taine's ipest, The Greatest Game. biological tion novel, and The Purple Sepphire, involving asuperciliztion ofthepast, Taine’s stories tiefantatic narratives toframeworks oforginal and logical scientie concepts. Speculation isoften profound onsuch questions asthe nature oftime, concept ofentropy cyclical Universes, ee. 4contemporary ilustrations. v-+32pp. 5% x834 21180.0 Paperbound $250 Seven Scizyce Fiction Novets, i. G. Welle Full unabridged texts of7wiencefiction novels ofthe master. Ranging from biology. physics, chemist, astronomy, tosociology and other stadies, Mr. Wells extrapolates whole works ofstrange and intriguing, character, “One will have togofartomatch this for entertainment, excitement, and sheer Pleanire "New York Times. Contents! The Time Machine, The Island of br Moreau, ‘The First Men inthe Moon, The Invisible Man, The War ofthe Worlds, The Food ofthe Gods, In‘The Days ofthe Comet. 10150P. 5% X8. 20264-X Clothbound $5.00 28 Screncr Ficrion Sroxies oF H. G. Writs “Two full, unabridged novels, Men Like Gods and Star Begotten, plus 26shortstoriesbythemastersciencefictionveriterofalltime!Storiesofspace,time,invention, exploration, futuristic adventure. Partial contents: The Country oftheBlind,IntheAbyss,TheCrystalEgg,TheManWhoCouldWorkMiracle,AStory ofDays 10Come, The Empire oftheAnts, The Magic Shop, The Walley oftheSpiders,AStoryoftheStoneAge,UndertheKnife,SeeReiders, ste.Anindispensable collection forthe library ofanyone interested! inscience ftion adventure. x8pp. 534X8. 20265-8 Clothbound $5.00 “Tuner Martian Novas, Edger Rice Burroughs Complete, unabridged reprinting, inone volume, ofThuvia, Maid ofMars: CChesimen ofMars: The Master Mind ofMars. Hours ofscience-fiction adven: ture byamodern master oryteller. Rest inlage clear type foreasy reading. 16illustrations byJAllen St.John. vi+}490pp. 534x8¥5 20089-6 Paperbound $2.50 [AN INTELLECTUAL AND CULTURAL Hisrony oFrite Westen Wort, Harry Elmer Barnet Monumental volume survey ofintellectual development ofEurope from Primitive culiores tothe presnt day. Every signifant product ofuman Intellect raced through history: ar, literature, mathematics, physical sciences,medicine,musi,technology, socalscience,religions,jurisprudence education,te. Presentation isIueid and specific, anaiysing indeait specific discoveries, theories, trary works, and s0\on. Revised (196s) byTecogoied scholars in specialized Gelds under thedirection ofProt. Bares Revised bibliography, Indexes. 24illustrations. Total ofxxix 4.1g34pp. 21275-0, 212769, 21277-7 "hie volume set,paperbound $835 CATALOGUE OF DOVER BOOKS “THe Merwovs oFEvics, Henry Sidguick Propounding noorganized system ofitsown, study subjects every major methodological approach toethics torigorous, objective analysis. Study dis Cusses and relates ethical thought ofPlato, Aristotle, Bentham, Clarke, Buler, Hobbes, Hume, Mill, Spencers Kant, and dozens ofothers. sidgwik retains Conclusions from each system which follow from ethical premises, rejecting thefaulty, Considered bymany inthefield tobeamong. themost important treatises onethical philosophy, Appendix, Index. xlvit 4.528pp. 5% %814. 21608:% Paperbound $250 ‘Tavroste Myrotocy, Jekob Grimm ‘Amilestone inWesten culture; the work which established onamodern hhasis thestudy ofhistory ofreligions and comparative religions. 4-volume work asembles and interprets everything available onreligious and folk Ioristic beliefs ofGermanic people {including Scandinavians, Anglo-Saxons, tte). Asembling material from such sources asTacitus, surviving OldNorse nd Icelandic texts, archeological remains, folktales, surviving superstitions, Comparative traditions, linguistic analysis, ete,Grimm explores pagan deities, heroes, folklore ofnature, religious practices, and every other area ofpagan German belie, Tothisday, theunrivaled, definitive, exhaustive study. Trans- lated byJ.S.Stallybrass from 4th(+883) German edition. Indexes. Total of Insvit +}i887pp. 594 x814.216020,21608-9,21604-7,21605-5ourvolumese,paperbwund $11.0 ‘Tue 1CwiNe, translated byJames Legge Called “The Book ofChanges” inEnglish, this isone ofthe Five Classics cecdited byConfucius, basic and central toChinese thought. Explains perhaps themost complex system ofdivination known, founded onthetheory that all things happening atany one time have characteristic features which can be isolated and related. Significant inOriental studies, inhistory ofreligions and Philosophy, and also'to Jungian psychoanalysis and other areas ofmodern European thought. Index. Appendixes. 6plates xxii-4-448pp. 536%814. 21062-6 Paperbound $2.75 History oF ANcteNr Puitosorny, W. Windelband One oftheclearest, most accurate comprehensive surveys ofGreek and Roman philosophy. Discusses ancient philosophy ingeneral, intellectual lifeinGreece fnthe7th and 6th centuries B.C., Thales, Anaximander, Anaximenes, Herac lita, the Eleatics, Empedocles, Anaxagoras, Leucippus, the Pythagorcans, the Sophists, Socrates, Democritus (20page), Plato (gopages), Aristotle (70pages), thePeripatetic, Stoic, Epicureans, Sceptics, Neo-platonists, Christan Apolo sist, etc. and German edition translated byHE. Cushman. xv++S08pp- 54x 8 20357-8 Paperbound $2.35 Tur Patace. oFPurasuar, Willian Painter Elizabethan versions of Italian and French novels {rom The Decameron, Ginthio, Straparola, Queen Margaret ofNavarre, and other continental sources =thevery work that provided Shakespeare and dozens ofhiscontemporaries‘withmanyoftheirplotsandsulyplotsand,therefore,justlyconsidered oneofthemost influential books inallEnglish literature. Itisalso abook that any reader will sill enjoy. Total ofevil 1.224pp.2160-8,21692-6,21699-4‘Threevolumeset,paperbound $6.75 | CATALOGUE OFDOVERBOOKS CATALOGUE OF DOVER BOOKS La BoneMe sy Giacomo Puccint, translated and introduced byEllen H.Bleiler ‘Complete handbook fortheoperagoer, with everything needed forfullenjoy- ment except themusical score itself, Complete Talian libretto, with new, ‘modern English line-by-line translation—the only libretto printing allrepeats: biography ofPuccini; thelibrettsts; background totheopera, Murger's La Boheme, ete; circumstances ofcomposition and performances; plot summary; andpictorial section of73illustrations showing Puccini, famous singers and performances, etc, Large clear type foreasy reading. 124pp. 5%x814. 20404.9 Paperbound $1.25 ANTONIO STRADIVARI: His LiFe AND Wonk (1644-1737), W.Henry Hill, Arthur F.Hill, and Alfred E,Hill Still theonly book that really delves into lifeand artoftheincomparable Italian craftsman, maker ofthefinest musical instruments intheworld today. ‘The authors, expert violin-makers themselves, discuss Stradivaris ancestry, his construction’ and finishing techniques, distinguished characteristics ofmany ofhisinstruments andtheir locations. Included, too,isstory ofintroduction ofhisinstruments into France, England, first revelation oftheir supreme ‘merit, and information onhislabels, number ofinstruments made, prices, ‘mystery ofingredients ofhisvarnish, tone ofpre-1684 Stradivari violin and changes between 1684 and 1690. Anextremely interesting, informative account forallmusic lovers, from craftsman toconcert-goer. Republication oforiginal (1902) edition. New introduction bySydney Beck, Head ofRare Book and ‘Manuscript Collections, Music Division, New York Public Library. Analytical index byRembert Wurlitzer. Appendixes. 68illustrations. gofull-page plates. 4incolor. xxvi +gi5pp. 5%x814. 20495-1 Paperbound $2.25, MusicaL Autockarus From MONTEVERDI To HinpEMiTH, Emanuel Winternitz Forbeauty, forintrinsic interest, forperspective on-the composer's personality, forsubtleties ofphrasing, shading, emphasis indicated inthe autograph but suppressed intheprinted score, themss. ofmusical composition arefascinating documents which repay close study inmany different ways. This 2-volume ‘work reprints facsimiles ofmss. byvircually every major composer, and many ‘minor figures—196 examples inall.Afull text points outwhat canbelearned from mss., analyzes each sample. Index. Bibliography. 18figures. 196plates. ‘Total ofi7opp. oftext. 774 %1054 21312-9,21313-7 Two volume set,paperbound $5.00 J.8. Bacn, Albert Schweitzer One ofthe few great fullength studies ofBach's life and work, and the study upon which Schweitzer’s renown asamusicologist rests, Onfirst appear- ance (1911), revolutionized Bach performance. The only writer onBach to ‘vemusicologist, performing musician, and student ofhistory, theology and philosophy, Schweitzer contributes particularly fullsections onhistory ofGer- ‘man Protestant church music, theories onmotivie pictorial representations invocal music, and. practical suggestions forperformance. Translated by Ernest Newman. Indexes, 5illustrations. 650musical examples, Total ofxix +928pp. 5%x814. 2681-4, 21682-2 Two volume set,paperbound $4.50 CATALOGUE OF DOVER BOOKS ‘Tue Principtes ov Psycuotocy, William James ‘The full Iong-course, unabridged, ofone ofthegreat clastics ofWestern literature and science. Wonderfully lucid descriptions ofhuman mental activity, thestream ofthought, consciousness, time perception, memory, imag- ination, emotions, reason, abnormal phenomena, and similar topics. Original ‘contributions areintegrated with thework ofsuch men asBerkeley, Binet, Mills, Darwin, Hume, Kant, Royce, Schopenhauer, Spinoza, Locke, Descartes, Galton, Wundt, Lotze, Herbart, Fechner, and scores ofothers. All contrasting interpretations ofmental phenomena areexamined indetail—introspective analysis, philosophical interpretation, andexperimental research. “Aclassic.” Journal ofConsulting Psychology, "The main lines are asvalid asever,” Prychoanalytical Quarterly. "Standard reading ...2classic ofinterpretation,” Paychiatric Quarterly. 94illustrations. 1408pp. 53 x8.20881-6,20382-4 ‘Twovolumeset,paperboundl $6.00 ‘Visuat ILLUsions: THEIR CAUSES, CHARACTERISTICS AND APPLICATIONS, M. Luckicsh “Seeing isdeceiving.” asserts theauthor ofthisintroduction tovirtually every type ofoptical illusion known. The text both describes and explains the principles involved incolor illusions, figure-ground, distance illusions, etc. 100photographs, drawings anddiagrams prove how easy itisto fool thesense:<irclesthataren'tround,parallelLinesthatseemtobend,stationary figuresthat‘em Yomove asyou state atthem —illustration after illustration strains our credulity atwhat wesee. Fascinating book from many points ofview, from applications forartists, incamouflage, etc. t0thepsychology ofvision. New introduction byWilliam Itleson, Dept. ofPsychology, Queens College. Index. Bibliography. xxi4z52pp. 534x814. 21580-X Paperbound $1.50 ' FAs AND FALLACIES INTHE NAME OFSCIENCE, 1 Martin Gardner ‘This isthe standard account ofvarious cults, quack systems, and delusions ‘which have masqueraded asscience: hollow earth fanatics, Reich and orgone sex energy, dianetics, Atlantis, multiple moons, Forteanism, flying saucers, ‘medical fallacies like iridiagnosis, zone therapy, etc. Anew chapter has been added onBridey Murphy, psionics, and other recent manifestations inthis field. This isafair, reasoned appraisal ofeccentric theory which provides ‘excellent inoculation against cleverly masked nonsense. “Should betead by everyone, scientist and non-scientist alike,” R.T.Birge, Prof. Emeritus of Physics, Univ. ofCalifornia; Former President, American Physical Society Index. x+365pp. 5%x8. 2030-8 Paperbound $2.00 ILLUSIONS AND DELUSIONS OF THE SUPERNATURAL AND THE OccuLT, D.H. Rawcliffe Holds uptorational examination hundreds ofpersistent delusions including ‘erystal gazing, automatic writing, table turning, mediumistic trances, mentalhealing,stigmata,lycanthropy, liveburial,theIndianRopeTrick,spiritualism,dowsing, telepathy, clairvoyance, ghosts, ESP, etc. The author explains and ‘exposes themental and physical deceptions involved, making this not only anexposé ofsupernatural phenomena, but avaluable exposition ofchar acteristic types ofabnormal psychology. Originally titled “The Psychology oftheOccult.”14illustrations. Index.551pP.5%*8.20508-7Paperbound $3.50 CATALOGUE OF DOVER BOOKS Fatty TALE Cottrcrions, edited byAndrew LangAndrewLang'sfairytalecollections makeuptherichestshelffulloftraditiona:children’s stories anywhere available. Lang supervised thetranslation ofstories from allover theworld—familiar European tales collected byGrimm, animal stories from Negro Africa, myths ofprimitive Australia, stories from Russia, Hungary, Iceland, Japan, and many other countries. Lang's selection oftrans lations areunusually high; many authorities consider that themost familiar tales find their best versions inthese volumes. All collections are richly deco- rated and illustrated byH.J.Ford and other artists, ‘THE BLUE Fatty Book. $7stories. 198illustrations. x4.s90pp. 534 814 21497-0 Paperbound $1.95 ‘Tux Gaxen Fary Box. 42stories. 100illustrations. xili +$66pp. 53% x8% 21439-7 Paperbound $1.75 ‘THe Brown Fairy Book. 92stories. 50illustrations, 8incolor. xii+ S50pP. 5%XBY. 21488-9 Paperbound $1.95 ‘Tue Best TALES oFHorFMANN, edited byE.F.Bleiler tostories byE.T.A.Hoffmann, one ofthegreatest ofallwriters offantasy. ‘The tales include “The Golden Flower Pot,” “Automata,” “A New Year's Eve Adventure,” “Nutcracker and theKing ofMice,” “Sand-Man,” and others. ‘Vigorous characterizations ofhighly eccentric personalities, remarkably imagi- native situations, and intensely fast pacing has made these tales popular all over theworld for150years. Editor's introduction. 7drawings byHoffmann.xxxili+419pP.594%BY. 181798.0Paperbound $2.25 Guost ano Honnor Stoniss oF Aupnose BIERcE; edited byE.F.Bleiler Morbid, eerie, horrifying tales ofpossessed poets, shabby aristocrats, revived corpses, and haunted malefactors. Widely acknowledged asthe best oftheir kind between Poe and the moderns, reflecting their author's inner torment and bitter view oflife. Includes “Damned Thing,” “The Middle Toe ofthe Right Foot,” “The Eyes ofthe Panther,” “Visions ofthe Night,” Moxon's Master,” and over adozen others. Editor's introduction. xxii +199pp. 5% x8%. 20767°6 Paperbound $1.50 ‘Tumex Gormic Novers, edited by E.F.Bleiler Originators ofthestill popular Gothic novel form, influential inushering in carly 19th-century Romanticism, Horace Walpole's Castle ofOtranto, William Beckford's Vathek, John Polidori’s The Vampyre, and aFragment byLord Byron areenjoyable asexciting reading orasdocuments inthehistory of English literature, Editor’s introduction. xi+ag1pp. 5% x844 21282-7 Paperbound $2.00 Brsr Gost Sronies oFLerAnu, edited byE.F.Bleiler ‘Though admired bysuch critics asV..Pritchett, Charles Dickens and Henry James, ghost stories bythe Irish novelist Joseph Sheridan LeFanu_ have hhever become aswidely known a8hisdetective fiction. About half ofthe16 stories inthis collection have never before been available inAmerica, Collec: tion includes “Carmilla” (pethaps thebest vampire story ever written), “The Haunted Baronet,” “The Fortunes ofSirRobert Ardagh,” and theclastic “Green Tea." Editor's introduction. 7contemporary illustrations. Portrait of LeFanu. xii+467pp. 5%X8. 20415-4 Paperbound $2.50 CATALOGUE OF DOVER BOOKS EASY-To-D0 ENTERTAINMENTS AND DIVERSIONS WITH Corns, CARDS, StRiNG, PAPER AND MatcHtEs, R.M. Abraham ‘Over gootricks, games and puzzles will provide young readers with absorbing fun. Sections oncard games; paper-folding; tricks with coins, matches and pieces ofstring: games fortheagile; toy-making from common household ‘objects; mathematical recreations; and. gomiscellaneous pastimes. Anyone inchargeofgroupsofyoungsters, includinghard-pressed parents,andinneedofsuggestions onhow tokeep children sensibly amused and quietly contentwillfindthisbookindispensable. Clearsimpletext,copiousnumberofdelight- fulline drawings and illustrative diagrams. Originally titled "Winter Nights Entertainments.” Introduction byLord Baden Powell. $29 illustrations. v-+ BOPP. 5Y%6 xB14. '20021-0 Paperbound $1.00 [AN Intropuction 10Citess Moves AND Tactics SIMPLY EXPLAINED, Leonard Barden Beginner's introduction totheroyal. game. Names, possible moves ofthe pieces, definitions ofessential terms, howgames arewon,etc.explained in '‘go-odd pages. With thisbackground you'll beable tositright down andplay Balance ofbook teaches strategy—openings,middlegame,typicalendgame play, and suggestions forimproving your game. Asample game isfully analyzed. True middle-level introduction, teaching youalltheessentials with- ‘out oversimplifying orlosing you inamaze ofdetail. 58figures. oapp. 3%x814. 21210-6 Paperbound $1.35 LAskER's MANUAL OFCes, Dr. Emanuel Lasker Probably thegreatest chess player ofmodern times, Dr. Emanuel Lasker held theworld championship 28Years, independent ofpassing schools otfashions, ‘This unmatched study ofthegame, chiefly forintermediate toskilled players, analyzes basic methods, combinations, position play, theaesthetics ofchess, dozens ofdifferent openings, etc., with constant reference togreat modem games, Contains abrilliant exposition ofSteinite’s important theories. Tntro- duction byFred Reinfeld. Tables ofLasker's tournament record. indices. ‘308diagrams. 1photograph. xxx +$49PP. 5%*#20640-8Paperhound $2.50 Commnattons: Tix Hxanr oFCites, Irving Chernev ‘Step-by-step from simple combinations to’complex, this book, byawel: Known chess writer, shows youtheintricacies ofpins, counter-pins, knight forks, and smothered mates. Other chapters show alternate lines ofplay t0 those taken inactual championship games; boomerang combinations: clasic ‘examples ofbrilliant combination play byNimzovich, Rubinstein, Tarrasch, Botvinnik, Alekhine and Capablanca, Index. 356 diagrams. ix’+245pp. 5%x84. 21744-2 Paperbound $2.00 How ro Souve Curss Prosirss, K.8.Howard Full ofpractical suggestions forthefanorthebeginner —who knows only the moves ofthechessmen. Contains preliminary section and 58two-move, 46 three-move, and &four-move problems composed hy27outstanding American problem creators inthelastgoyears. Explanation ofallterms and exhaustive Index. “Just what iswanted forthestudent,” Brian Harley. 112problems, solutions. vi+171pp. 5%X8. 20748-X Paperbound $1.50 ! CATALOGUE OF DOVER BOOKS Sociat. THoUcHT FROM LonE vo SclENCE, HE. Barnes and H.Becker An immense survey ofsociological thought and ways ofviewing, studying planning, and reforming society from earliest times tothepresent. Includes Thought onsociety ofpreliterate peoples, ancient non-Western cultures, and ‘every great movement inEurope, America, and modern Japan. Analyzes buns dreds ofgreat thinkers: Plato, Augustine, Bodin, Vico, Montesquieu, Herder, Comte, Marx, ec. Weighs thecontributions ofutopians,sophists, fasts and ‘communists; economists, jurists, philosophers, ecclesiastics, and every 1th land oth century school ofscientific sociology, anthropology, and social psy- ology throughout theworld, Combines topical, chronologteal,andregional approaches, treating theevolution ofsocial thought asaprocess rather than 4as.a series ofmere topies."Impremive accuracy, competence, and discrimina. tion «easly the ber single survey.” Nation. ‘Thoroughly revised, with new material up(01960. 2indexes. Over 2200 bibliographical notes. Three volume set. Total of1580p. 534 x8 1205016, 20902-4 20003.2 ree volume set,paperhound $9.00 AHistory oFHistonicaL Watrine, Harry Elmer Barnes Virtually theonly adequate survey ofthewhole course ofhistorical witing in-a single volume. Surveys developments from thebeginnings ofhistoriog: raphy intheancient Near East and theClassical World, upthrough the Cold’ War. Covers major historians in.detail, shows interclationship. with cultural background, makes clear individual contributions, evaluates ‘and estimates importance: also enormously rich upon minor authors and thinkers‘whoareusuallypassedover.Packedwithscholarship andlearning,clear,eailyveriten, Indispensable toevery student ofhistory. Revised and enlarged up{01961.Indexandbibliography. x0-+442pp.534X81420104-X Paperbound $2.75 JOUANN SemAsTIAN Bacit, Philipp Spit ‘The complete and unabridged text ofthedefinitive study ofBach. Written some 7oyears ago, itis still unsurpassed foritscoverage ofnearly allaxpects ofBach's lifeand work. There could hardly beafiner non-technical introduc: tion toBach's music than thedetailed, Iucid analyses which Spitta provides forhundreds ofindividual pieces. 26slid pages aredevoted totheBminor mass, forexample, and opages t0the glorious St,Matthew Passion, ‘This monumental set also incldesamajoranalysisofthemusiofthes8thcentury: Buxtchude, Pachelbel, etc. "Unchallenged 2:the last word onone ofte supreme geniuses ofmusic,” John Barkhamn, Saturday Review Syndicate, Total ofaB1gpp. Heavy cloth binding. 594x8. 222780, 22279.9 Two volume set,clothbound $15.00 BEETHOVEN AND His Nine Syatewontes, George Grove Inthismodern middle-level classic ofmusicology Grove notonly analyzes all nine of‘Beethoven's symphonies very thoroughly interms oftheir musical structure, butalo discuses theciteumastances under which they were written, Beethoven's stylistic development, and much other background material. Thisisanextremelyrichbook,yetveryeasilyfollowed;itishighlyrecommendedtoanyone seriously interested in’music. Over 250 musical passages. Index. vill 4O7pp. 5%x& 20884-4 Paperbound $2.25 CATALOGUE OF DOVER BOOKS y's FUN To MAKE. THINGS FROM ScRAP MATERIALS, Evelyn Glantz Hershoff ‘What useareempty spools, tincans, bottle tops? What can bemade from rubber bands, clothes pins, paper clips, and buttons? This book provides simply worded instructions and large diagrams showing you how tomake cookie cutters, toytrucks, paper turkeys, Halloween masks, telephone sets, aprons, linoleum block- and spatter prints—inallgg9projects!Manyareeasy enough foryoung children tofigure out forthemselves; some challenging ‘enough Coentertain adults; allareremarkably ingenious ways tomake things from materials that cost pennies orless! Formerly "Scrap Fun forEveryone.” Index. 214illustrations. 373pp. 5% X814. 212513 Paperbound $1.75 SyMBoLic Looic and Tur Game or Loctc, Lewis Carroll “symbolic Logic” isnotconcerned with modern symbolic logic, butisinstead1collection ofover380problemsposedwithcharmandimagination, usingthesyllogism andafascinating diagrammatic method ofdrawing conclusions. In“The Game ofLogic” Carroll's whimsical imagination devises alogical game j played with 2diagrams andcounters (included) tomanipulate hundreds of J tricky syllogisms, The final section, “Hit orMiss” isalagniappe of101addi- tional puzzles inthedelightful Carroll manner. Until this reprint edition, both ofthese books were rarities costing upto$15 each. Symbolic Logic:Index.xxxi-}1g9pp.TheGameofLogic:g6pp.2vols.boundasone.534x8.2092-8 Paperbound $2.50 MATHEMATICAL Puzzits OFSAM Lovo, PART! selected and edited byM.Gardner ‘Choice puzzles bythegreatest American puzzle creator and innovator. Selected from hisfamous collection, “Cyclopedia ofPuzzles,” they retain theuniquestyleandhistoricalflavoroftheoriginals.Thereareposersbasedonarithmetic, algebra, probability, game theory, route tracing, topology, counter and sliding block, operations research, geometrical dissection, Includes thefamous “14-15” puzzle which wasanational craze, and his“Horse ofaDifferent Color” which fold millions ofcopies. 117ofhismost ingenious puzzles inall.120line drawings and diagrams, Solutions, Selected references. xx-}167pP. 5%X8. 20498-7 Paperbound $1.35 Smmie Ficurs Axo How toMake THEM, Caroline Furness Jayne 107string figures plus variations selected from thebest primitive and modern examples developed byNavajo, Apache, pygmies ofAfrica, Eskimo, inEurope,‘Australia, China,etc.Themostreadilyunderstandable, easy-to-follow bookinEnglish onperennially popular recreation. Crystal-clear exposition; step-by: step diagrams. Everyone from kindergarten children toadults looking for ‘unusual diversion will beendlessly amused. Index. Bibliography. Introduction byA.C, Haddon. 17full-page plates,g6oillustrations. xxiii+4o1pp.5%4x814.Q0152.X" Paperbound $2.25, Papen FOLDING FoR BEcINNERS, W.D.Murray and F.J.Rigney Adelightful introduction tothe varied and entertaining Japanese artof ‘origami (paper folding), with afull, crystal-clear text that anticipates every difficulty: over 275clearly labeled diagrams ofallimportant stages increation ‘You getresults ateach stage, since complex figures are logically developed from simpler ones. 43difterent pieces areexplained: sailboats, frogs, roosters, ‘etc. 6 photographic plates. 279 diagrams. o5pp. 536 x834.pm " '20713°7Paperbound $1.00 | CATALOGUE OF DOVER BOOKS Paincirurs oF AKT Hisrony, i.Wolptin Analyzing such terms as“baroque,” “clawic:” “neoclasic” “primitive” “picturemyve” and 164diferent works byartists like Botticelli, van Cleve, Diver, Hobiema, Holbein, Hals, Rembrandt, Titian, Brueghel, Vermeer, and ‘many others, theauthor establishes theclassifications ofarthistory and syle onafirm, concrete basis, ‘This classic ofart criticism shows, what realy ‘occurred between the 14th-centary primitives and thesophistication ofthe ‘th century interms ofbasic attudes and philowophies. "A.remarkable lesson intheartofacing,” Sat Rev. ofLiterature, ‘Translated from the 7thGermanedition.150illustrations. s54pp.64*934.20276:Paperbound $225, Paincrive nr,rantBoas“This authoritative and exhaustive work byagreat American anthropologist covers the entre gamut ofprimitive art. Pottery, leatherwork, metal work, Stone work, wood, basketry, aretreated indetail “Theoris ofprimitive ar historical depth inarthistory, technical vrtdosty, unconscious Tevels ofpat teming, symbolism, styles, literature, music, dance, et. Amust book forthe interested Iayman, theanthropologist, artist, handieratter (hundreds ofun ‘sual moti), andthe historian. Over goo ilustrations (so ceramic vessels,{2totempoles.ete).$760P.5368 20025.6Paperhound $2350 ‘Tite Gewturwan axp Ganiner Maxen’s Dintcron, Thomas Chippendale Areprint ofthe #762 catalogue offurniture designs that went ontoinfluence generations ofEnglish and Colonial and Early Republic American furniture Imakers. The soo plat, most ofthem full-page sized, show. Chippendales designs for French (Louis XV), Gothic, and. Chinese-manner chairs, sla, ‘canopy and dome beds, cornices, chamber organs, cabinets, shaving tales,commodes,pictureframes,frets,candlestands,chimncypieces,decorations, ete‘The drawings are allelegant and highly detailed: many include construction diagrams and elevations. Asupplement of24photographs shows surviving Picee oforiginal and Chippendale-ryle picces offurniture. Brief biography ofChippendale byN.1.Bienenstock, editor ofFurniture World. Reproduced from the 1762 edition, a0 plates, plus 19photographic plates. vi+}249PP. oxy, 21601-2 Paperbound $3.50 Avenicas Anriqut FURNITURE: ABoox FoR AMATEURS, Edgar €.Miller, Jr. Standard introduction and practical guide toidentification ofvaluable ‘American antique farniture. 115 illustrations, mostly photographs taken by theauthor in148private homes, arearranged inchronological order inexten- sive chapters onchairs, sofa, chest, desks, besteads, mirrors, tables, clocks, sand other articles, Focus ison furnivure accesible tothecollector, including fimmplr picces and alarger than usual coverage ofEmpire syle. Introductory chapters identify sructaral elements, characteristics ofvarious stjles, how © void fakes, ete) “We are frequently asked toname some book onAmerican furniture that will meet therequirements ofthenovice collector, thebeginningdealer,and.thegenctalpublic... WebelieveMr.Miller'stwo‘volumes more completely satisfy this specification than any other work,” “Antiques: Appendix: Indes. Totalofvi}s100P.7%4X1096 21500-7,216004 Two volume set,paperbound! $7.50 CATALOGUE OF DOVER BOOKS THE BAb Cxtt0's Book oFBrasts, Mone BFAsTs FoR Worse CHILDREN, and AMonat Atewanrt, H. Belloc Hardly and anthology ofhumorous verse has appeared inthe last 50years without atleast acouple ofthese famous nonsense verses, But one must see the entire volumes—withallthedelightfuloriginalillustrations bySirBasil Blackwood—toappreciate fully Belloc's charming and witty verses that play sosubacidly ontheplatitudes oflifeand morals that beset hisday —and ours,Agreathumorclassic.Threebooksinone.Totalof157pp.5%x8.20749-8 Paperbound $1.00 ‘Tue Devin's Dictionary, Ambrose Bierce Sandonic and irreverent barbs puncturing the pomposities and absurdities of American politics, business, religion, literature, and arts, bythe country’s greatest satirist intheclassic tradition, Epigrammatic asShaw, piercing as Swift, American asMark Twain, Will Rogers, and Fred Allen, Bierce will always remain the favorite ofasmall coterie ofenthusiasts, and ofwriters and speakers whom hesupplies with “some ofthe most gorgeous witticisms 7 oftheEnglish language” (H.L,Mencken). Over 1000entries inalphabetical order. 144PP. 5%*8 20487-1 Paperbound $1.00 Tur Compete Nonsense oF EDWARD LEAR. ‘This istheonly complete edition ofthis master ofgentle madness available atapopular price. ABook ofNonsense, Nonsense Songs, More Nonsense Songs and Stories intheir entirety with alltheoldfavorites that have delighted children and adults foryears, The Dong With ALuminous Nose, The Jumblies, ‘The Owl and thePussycat, and hundreds ofother bitsofwonderful nonsense. 214limericks, 3setsofNonsense Botany, 5Nonsense Alphabets, 546drawings byLear himself, and much more. g2opp. 5%x8,20167-8 Paperbound $1.75 Tue Wir AND HuMon oFOscar Wipe, ed. byAlvin Redman Wilde athis most brilliant, in1000 epigrams exposing weaknesses and hypocrisiesof“civilized”society.Dividedinto49categories~sin, wealth,women, America, ete.—to aidwriters, speakers. Includes excerpts from histrials, books, plays, criticism. Formerly “The Epigrams ofOscar Wilde.” Introduction by \Vyvyan Holland, Wilde's only living son. Introductory essay byeditor. s6opp. BH XB. 20602-5 Paperbound $1.50, ACuiito’s Printer oF NATURAL History, Oliver Herford Scarcely ananthology ofwhimsy and humor has appeared inthelast 50years without acontribution from Oliver Herford. Yet the works from which these ‘examples aredrawn have heen almost impossible toobtain! Here atlast are Herford’s improbable definitionsofamenagerieoffamiliarandweirdanimals, each verse illustrated bytheauthor's own drawings. 24drawings in2colors: 24additional drawings. vii+g5pp. 614x6. 21647-0 Paperbound $1.00 ‘Tae Browntss: THxiR Boox, Palmer Cox ‘The book that made the Brownies ahouschold word, Generations ofreaders have enjoyed theantics, predicaments and adventures ofthese jovial sprites,whoemergefromtheforestatnighttoplayortocometotheaidofadeservinghuman. Delightful illustrations bythe author decorate nearly every page. 24short verse tales with 266 illustrations. 155pp. 694X944. 21265-8 Paperbound $1.50 | CATALOGUE OF DOVER BOOKS ‘Tue WoxpenruL Wizano oF Oz, L.F,Baum AAtheoriginal W.W.Denslow illustrations infullcolor—as much apart of “The Wizard” asTenniel's drawings areof"Alice inWonderland.” “The Wieara” istillAmerica's best-loved fary tale, inwhich, astheauthor expresses, it,“The wonderment and joyareretained and theheartaches and nightmares leftout” Now today's young readers canenjoy every word and wonderful pic ture oftheoriginal took. New introduction byMartin Gardner. ABaum bibliography. 2§full-page color plates. vil+p268pp. 5%*8. B0691-2 Paperbound $1.95 “The Manvetous LAND oFO2, L.F.Baum “This istheequally enchanting sequel tothe“Wizard,” continuing theadven- tures ofthe Scarecrow and the Tin Woodman. The hero this time isalite toynamed Tip, andallthedelightful Ormagic issill present. This fsthe Orbook with theAnimated Saw-Horse, theWoggle-Bug, and Jack Pumpkin head. Alltheoriginal John R.Neill illustrations, x0.infull color. a7pp. x8. ‘20602:0 Paperbound $175 Auice’s Aovewrunes UnoEx Grounp, Lewis Carroll “The original Alice inWonderland, hand-lettered and illustrated byCarroll himseif, and originally presented ataChrstinas gift toachild-riend. Adults aswell aschildren will enjoy thischarming volume, reproduced faithfully inthis Dover edition, While thestory isessentially thesame, there arealight changes, and Carroll's sprtely drawings present anintriguing. alternative to thefamous Tennielillttrations, One ofthemost popular books inDover's Catalogue. Introduction byMartin Gardner, 38illustrations. 128pp. 53*8Y4- 214826 Paperbound $1.00 ‘Tue Nutseny “Attee,” Lewis Carroll While most ofusconsider Alice inWonderland astory for children ofall ages, Carroll himself feltitwas beyond younger children. Hetherefore pro vided this simplified version, iMlustated with the famous Tenniel drawings enlarged and colored indelicate tints, forchildren aged “trom Nought to Five" Dover's edition ofthis now rare classic isafaithful copy ofthe 1889 printing, including 0ilustrations by‘Tenniel, and front and back covers Feproduced infull color. Intoduction byMattin Gardner. xxii -+yp. 54x9% 21610-1 Paperbound $175 ‘TueSronyoFKineAxrWuxANDHisKytcitts,HowardPyle Afast-paced, excitingretellingofthebestknownArthurianlegendsforyoungTeaders byone ofAmerica’s best story tellers and illustrators, The sword Excalibur, wooing ofGuinevere, Merlin and hisdownfall, adventures ofSit Pellias and Gawaine, and others. The pen and ink llustraions. are vividly imagined and wonderfully drawn. 41illustrations. xvit +s13pp. 614 %9¥4- 21451 Paperbound. $2.00 Prices subject tochange without notice, Available atyour book dealer orwrite for free catalogue toDept. Adxi, Dover Publications, Ine, 1Varick St, NL.,N'Y. 10014. Dover publishes more than 150books each year onscience, elementary and advanced mathematics, Diology, music, art, Iiterary history, social siences and other areas