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Table Errata Source: Mathematics of Computation, Vol. 41, No. 164 (Oct., 1983), pp. 775-783 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2007718 Accessed: 19/04/2010 16:53 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Mathematics of Computation. http://www.jstor.org TABLE ERRATA 594.-W. MAGNUS, F. OBERHETTINGER & R. P. SONI, Formulas and Theorems for the Special Functions of Mathematical Physics, third enlarged edition, Springer, New York, 1966. tU tn p. 1: Line 6 up: For ! a read p. 13: Line 11 up: Read 4(1 + z) = instead of = Add,IzI< 1, p. 13: Line 10 up: Read 4(1 + z) = instead of = Add,O< lzl <1. p. 13: Line 6 up: For +(z), read +(x). p. 14: Line 1 up: For ;n(Z + 1/m), read 4("i)(z + 1/m). p. 15: Line 2 up: For n( '(), read 4 (n)( I ). p. 21: Line 6: For Re z > -1, read Re z > 1 [cf. EH I 32(7)]. p. 21: Line 7: For UZ, read tz [cf. EH I 32(8)]. p. 27: Line 7: Add , n = 1,2,.. [cf. the first formula on this page]. p. 32:Line 3: Addn= 1,2. Line 10: For Et+ ', read E(m? '); for Et, read E p. 33: Last line: For F(m + 2), read (m + I)F(m + 2). Add I log z < 2,r. p. 34:Line 2up: ForRes< 1,0<a< 1, read Re s < 0, 0 < a 1 [cf. EH I 28(6)]. p. 38: Line 1 1: For z2 F,, read Z2 F,. aa p. 71: Last line: For a read p. 74 } 3.3.3 Replace z by x everywhere. p. 75 p. 75: Line 7: AddIargzI< 2,r. p. 75: Line 6: up is incorrect. p. 88: Line 3 up: Add Re M > - 2, Re v > -. p. 89: Line 4: For J2,+2m+? (z), read J2v+2m+ 1(2z). p. 124: Line 10 up: For F(v + 1), read a-F(v + 1). p. 131: Line 8 up: Delete (-_)n -. Line 6 up: Move B2n to line 5 up. p. 132: Line 6: Replace x by m2. Line 13 up: For [(2m2, read [(2m. p. 154: Line 3: For z 2 )I, read (z + ) [cf. EH 1 141(17)]. p. 154: Line 7 up: For F(' - y), read F( - v). Line 4 up: For [z + (z2 - 1)1/2], read [z - (Z2 - 1)1/2]. p. 167: Line 9 up: For XF(O - v - 4tt, read XxF(4 'P - 2M- [cf. EH I 144(12)]. p. 174: Line 2 up: Replace z by x. 775 776 TABLE ERRATA p. 176: Line 6: For , read 2 [cf. GR 1019]. n=0 m=0 For (x), read (z). p. 177: cf. Hansen [1, p. 370] Formulas (56.2.8) and (56.2.9): Line 7: Delete the comma. Line 9: For I t j< 1, read I h I< 1. 00 Line 10: Add = : hnQn(x), -1 < x < 1, n=O Line 11: For X (z - t), read (z -t) p. 178: Line above 4.5: For Q, read Qg,. p. 179: Line 8: For Qm(cos 0), read Qm(cos 0') [cf. EH I 169(3) and Errata]. p. 180: Line 11 up: Replace the German $ 's by Latin P's. p. 182: Line 11 up: For (cos i), read (-cos i). p. 195: Lines 17 and 18 up: According to EH 1 162 Sec. 3.9.1: Read: ... in -,r < arg c < r even when the series does not converge, provided only that D is not a real number.... p. 197: Line 2 up: The asymptotic behavior should read ei7rg71/2(2z)-n-3/2F(n + 3 + tL)/F(n + 2). p. 200: Line 11 up: Forv -- 0, read v -- 0. p. 213: Line 8 up: For )3, read )3 X. This corrects the corresponding entry in MTE 575 (Math. Comp., v. 36, 1981, p. 316). p. 215: Line 12: Read + Q( fl) instead of -Q( fl) [cf. Szegb [2, (4.62.9)1]. Line 9 up: The integral representation is incorrect. For dz read p. 220: Line 4: Replace F(n + 1) by n!F(2XA). p. 222: The last two equations of the recurrence relations for CA" are incorrect. Read instead: C> (x) = xC,A+t(x) + n + 2X CX(x). p. 233: Line 8: Replace 1 - 2tcos(0 + p) + t2 l - 2tcos(0 - p) + t2 by 1 -2tcos(0 + qp) + t2 + 1 -2tcos(-q) + t2. According to Hansen [1, p. 306], we have tk Pk(cos x)Pk(cosy) u + vK u + v) - -K[-(t sin x sin y)l/2] -=--K[?(-t sin x sin y)I/2], where u = [1 + t2- 2tcos(x + y)]I/2, v = [1 + t2- 2tcos(x -y)]l/2. TABLE ERRATA 777 p. 234: Line 12 up: Read m = 1 instead of m = 0. p. 235: Line 9 up: Read i1- x2 instead of Vx2- 1. [Cf. the correction on p. 188 in MTE 477 (Math. Comp., v. 25, 1971, p. 201)]. p. 235: 5.4.3 Asymptotic expansions: Note that the convention of EH I is used here, according to which (X2 - 1)1/2 is defined by (x2 - 1)1/2 (x- 11)2(x + 1)/2 This means a deviation from the ordinary definition, i.e., w: = exp(a ln w). The cut ix E (-x, x) of (x2 - 1)' divides the complex x plane in two disconnected parts. In the right-half plane, i.e. Re x > 0, we have (indeed) (X2 - 1) = (x - 1)a(x + 1)a. However, in the left-half plane, Re x < 0, we have (x2 - = e-2ila(x -)a(x + 1)a Im x > 0, = e+2ira(x -)a(x + 1)a, Im x < 0. It may be noted, furthermore, that Nx+ 1 (x + 1)u for all x E C\(- x1]. By taking the appropriate limit of the right-hand side on part of the cut, (- x, -1), it follows that the discontinuity across this part of the cut vanishes. This implies that the analytic continuation of the right-hand side on (- oo, -1) is equal to the left-hand side. The remaining cut is [- 1, 1]. p. 238: Line 7: For 0 < r, 0 - m < v, read 0 ( < 0 < p. 239: Line 5: For cos n8, read cos m8 [Cf. EH I 168(2)]. p. 267: Line 6 up: For (c - a), read z(c - a). p. 271: Line 4: For W(w1, w6), read W(w,, W7). Line 5: For W(w,, W7), read W(w,, w6) [Cf. EH I, pp. 253 and 259]. p. 277: Line 6 up: For 2 -cez, read 21-ce2z. p. 284: Lines 2 and 3: Replace T(v - z) by (vp - z) In Line 2 the right-hand side should be preceded by a minus sign. p. 288: Line 11 up: For -2y, read +2y. [Cf. EH I 261(13)]. Line 2 up: For z Re c-2I read I z 2-Re c[Cf. EH I, pp. 257 and 261]. p. 289: Line 7: For -N, read -M. p. 304: Line 4: For z WK+ 1/2,-1/2(Z), read - z WK+ 1/2,-1/2(Z) p. 493: In the definition of n! replace the commas by the symbol-. 778 TABLE ERRATA The preceding list of errata is an abridgment of our Corrigenda Report 82 03. The inaccuracies and misprints of minor importance listed in this report have been omitted in this abridgement. Copies of the report will be available upon request. H. VAN HAERINGEN AND L. P. KOK Department of Mathematics Delft University of Technology Delft, The Netherlands Institute for Theoretical Physics P.O. Box 800 Groningen, The Netherlands 1. ELDON R. HANSEN, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, N.J., 1975. 2. G. SZEGO, Orthogonal Polvnomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1975. 595.-Bateman Manuscript Project: A. ERDtLYI, W. MAGNUS, F. OBERHETTINGER & F. G. TRICOMI, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, Toronto, London, 1953. p. 35: (1) For z < 2r, read 0 <Iz < 2,r. p. 136: (43) Second line: For 2,., read 2y. p. 143: Line 2: For (20), read (21). p. 143: Line 3: For F(4' -t), read 1(' - v). p. 144: (12) For z read x. p. 231: (5) For X, read = p. 262: (6) For Re c -2, read 2- Re c. p. 266: (17) For cl'X6, read e"'. 596.- Bateman Manuscript Project: A. ERDtLYI, W. MAGNUS, F. OBERHETTINGER & F. G. TRICOMI, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, Toronto, London, 1953. p. 64: (6) Read a-v(z)P-' instead of (2az)'-v. p. 82: (24) Read Im(a2z) < 0 instead of Im(a2z) > 0. p. 183: (47) Read cos mT instead of cos nT. p. 186: (30) Delete 1 + . p. 225: (5) Read x + 1 - n instead of x - n. (6) Add 0 < x < N, x integer. [Cf. G. Szego, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol 23, Amer. Math. Soc., Providence, R.I., 1975, Sec. 2.82.] 597.-Bateman Manuscript Project: A. ERDtLYI, W. MAGNUS, F. OBERHETTINGER & F. G. TRICOMI, Higher Transcendental Functions, Vol. III, McGraw-Hill, New York, Toronto, London, 1955. p. 101: (15) For 1 + 2A, read 1 - 2A. 598.-Bateman Manuscript Project: A. ERDtLYI, W. MAGNUS, F. OBERHETTINGER & F. G. TRICOMI, Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, Toronto, London, 1954. TABLE ERRATA 779 p. 10: (29) The result 0 given for m even is incorrect. For example, take m = 2, n -1. p. 11: (4) On the right-hand side y should be replaced by y/2 everywhere. p. 19: (5) Ready = 1,2,3,... instead of y = 0, 1,2,. p. 20: (11) Read 2 instead of 2 Orcr<(y-+4n)/2 O<r<(y-+-n)/2 p. 30: (8) The right-hand side should be divided by sinh(vy); hence replace cosh(ry) by ctnh(,Ty). p. 67: (38) The result 0 given for m odd is incorrect. For example, take m - 1, n - 1. p. 67: (38) For 0 m 2n read 0 < m 2n. p. 67: (38) For {cos read cos{ p. 68: (2) For 'v-' read -iv'-'. Add v # 0. p. 68: (4) On the right-hand side y should be replaced by y/2 everywhere. p. 68: (5) For 'B read -i=B. p. 70: (15) For gii read -ui. p. 70: (16) For 1 -ay read v- 1 -ay. p. 70: (17) For (- 1)n+ 1 read (- l). p. 73: (13) The right-hand side should read 'j tanh(vy). p. 120: (20) ForO<Re a<n,read0<ReA<n. p. 122: (32) Read 2 instead of -2, both in the first line and in the second line on the right-hand side. p. 308: (5) For arg(l - ab) read arg(1 + ab). p. 310: (22) For -7T < 0 < 7T, read 0 < 0 < 7T. p. 321: (37) is wrong, even after the correction suggested in the errata on page xvi. Its right-hand side should read (2a)Y2sP(s)exp( 4)cos(,rs)Ds(a 1/2), Res >0. 599.-Bateman Manuscript Project: A. ERDtLYI, W. MAGNUS, F. OBERHETTINGER & F. G. TRICOMI, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, Toronto, London, 1954. p. 199: (92) For aK, read a'+l. p. 278: (22) For r(M + n), read r( + n + 1). p. 284: (1) The right-hand side should be divided by n!. In MTE 533 it is stated that for F(a + n + 1) one should read r(a + 1). Unfor- tunately, this is incorrect. The reason for this mistake is that the incorrect formula (3) on page 284 has been used for the derivation. (Cf. the corresponding entry in our list.) p. 284: (3) The right-hand side should be multiplied by F(a + n + l)[n!T'(a + 1)1-1. 780 TABLE ERRATA The preceding list of errata is an abridgment of our Corrigenda Report 82 04. The inaccuracies and misprints of minor importance listed in this report have been omitted in this abridgment. Copies of the report will be available upon request. H. VAN HAERINGEN AND L. P. KOK Department of Mathematics Delft Unviersity of Technology Delft, The Netherlands Institute for Theoretical Physics P. 0. Box 800 Groningen, The Netherlands 600.-M. ABRAMOWITZ & I. A. STEGUN, Editors, Handbook of Mathematical Func- tions, Dover, New York, 1965. p. 336: 8.10.3 The right-hand side should read, in part, 7Te-i'ACsc[7r(v - t)]. 2F(l + M) p. 336: 8.10.6 In the denominator of the right-hand side, for F(' -7 y), read F(- + v). p. 440: 10.1.41 For (7/x), read x-'. 10.1.42 For (7/x), read x-'. 10.1.43 For (2X/x), read x-1. 10.1.44 For (2X/x), read -x-. p. 445: 10.2.35 Add Ire"Ia <IpP. p. 508: 13.5.6 For 6Ab - 2, read 2 -6b. p. 508: 13.5.9 For A(a), read A(a) + 2y. H. VAN HAERINGEN AND L. P. KOK Department of Mathematics Delft Unviersity of Technology Delft, The Netherlands Institute for Theoretical Physics P. 0. Box 800 Groningen, The Netherlands 601-I. S. GRADSHTEYN & I. M. RYZHIK, Table of Integrals, Series and Products, Corrected and enlarged edition, Academic Press, New York, First Printing, 1980. P. xxxii: Line 5 from bottom. Read J for j P. 354: 3.531.1; and p. 533: 4.233.2. One is the negative of the other. Except for sign, there are slight differences in the numerical values. P. 361: 3.551.5. Add f > y. P. 373: 3.631.15. The portion in braces can be expressed as 2-m 2 -M,- mr-n mr-n n -r2FIVm , 2 ;1 2 -2-m-1 (l+m)r(n2 ) co n-rn (rm+ n +) 2 TABLE ERRATA 781 P. 374: 3.631.17. The portion in braces can be expressed as - F(1 + )F( ' ) sin( ) IFm +n + i)2 P. 374: 3.631.18. The second parameter of the 2F, should read -(a + m)/2. The same error is in the source WA (Watson), but p. 313, not p. 342. It is also given incorrectly in MO (Magnus and Oberhettinger), p. 16. The result is also valid for a an integer. See also Errata # 282, Math. Comp., v. 14, 1960, p. 221. If q77 Lm =fcosm x cos ax dx, 0 L2r --2asinra rI (2kr) + (2r s 22r k=o 4(r - k)2- a2 22ra {2r+ I8 L2r+_ 2asina7Tr (2k 2 l 22r+I k=o (2r + 1 2k) a P. 374: 3.631.20. Add Re(n) > -1. P. 374: 3.632.3. Add the condition n > 0; Also for n s 0 and p not an integer fcosPxsin(p + 2n)xdx = 2FI(1- n,-p -n; I-p-n; (p + 1 - 2n)?Tcot(p -n) (p + 1-n)(l-n) for p an integer and n #0 O, use p. 373, 3.63.15; for p an integer, use p. 373, 3.631.16 if n = 0. P. 447: 3.824.3. The formula is correct, but an equivalent and more tractable form is 2 2m-l 2 (_1 (m -k e-2 . 22nla k=O ( k Note: This formula as well as p. 448: 3.824.5,.6,.7 can be expressed in hypergeometric form since 2 (_)Zk - r + ) _ +)2FI(-q, I; p -q + I; -Z). P. 448: 3.824.4. Lower limit of second sum should be 0. P. 449: 3.827.1. The condition on v should read 0 < Re v < 4. P. 455: 3.832.20. Add m = 0,1,2,. P. 455: 3.832.22. Read s > 0, n > 0, Re(a) > 0; for n = 1, see p. 406: 3.723.3. P. 459: 3.838.3-.4. These are both special cases of 3.838.2. The latter holds for non-integer values of the exponents. 782 TABLE ERRATA P. 477: 3.893.4. The integral is meaningless since the integrand has an infinite number of poles on the path of integration, unless a/b is an integer in which case it becomes equivalent to 3.893.5 or 3.893.6 according as a/b is odd or even, respectively. P. 478: 3.895.2,.3,.5,.6,.8, .10. Remove the restriction p #& 0 since all expres- sions are valid in the limit as p 0. P. 479: 3.895.12,.14. Read a ::O, n > 0. P. 479: 3.895.11, .13. Read a > 0. More, tractable forms for equations (11)-(14) are (11), (13): Re {22n?2 sin( a 2a )7T(n ? a + ) } (12),(14): RI422n?cos(m 2+if (+21+ 2) )} P. 481: 3.899.2: On the left-hand side read cos(4n + I)x for sin(4n + I)x. On the right-hand side read -(k/p )2 in the exponent. The latter mistake is also in the source BI. P. 1073: 9.524. On the right-hand side for k2 read kz. HENRY E. FETTIS 1885 California, Apt. 62 Mountain View, California 94041 P. 589: 4.389.5. The right-hand side should read +T [C + +(p + 1) -21n2] (p > -1). 2P?1 The same correction should be applied to the source BI. EMERIC DEUTSCH Department of Mathematics Polytechnic Institute of New York Brooklyn, New York 11201 The following errata were detected by use of the Soviet MIR-2 computer. The author thanks Y. L. Luke for his aid in preparing this note for publication. P. 298: 3.254.2. Replace IF(X - j) by F(X - 2v). P. 424: 3.768.3. Replace the right-hand side by a-'- 1/2S,+ 1/2,1/2(a) [Re v > -1] Note that this is a special case of an alternative representation for 3.768.5 given by H. van Haeringen and L. P. Kok in their list of errata. See Math. Comp., v. 39, 1982, pp. 747-757. TABLE ERRATA 783 P. 427: 3.771.12. In the first equation for sv,+ 1(au) read sv_-l,+ (au). In the second equation for (P + {) read (P + )-1. [Rev > - ]. P. 681: 6.551.1. Replace (P - ) + J,(Y)SA-1/2,(y) by (P - This pertains to the 4th edition only. 6551.1 is a special case of 6.561.13. See below. P. 683: 6.561.6. In the right-hand side divide the second term by 7. P. 684: 6.561.13. For the right-hand side read 2 + 1) + a -{(.t + v - l)J(a)SA_ (a)- J ,(a)S,bV(a)}. P - ,t + 1) This is equivalent to 5 a-A(fi + v - l)J(a)sA_,',_l(a) - J-l(a)sALP(a)1. P. 702: 6.592.2. In the second term of the right-hand side replace 2'-" by 2'- . See 6.592.7 below for a special case. P. 702: 6.592.7. Replace 2 sec by , 7 sec 2 P. 849: 7.512.6. The right-hand side should read B(X, / - X)(1 - z/b)-a. ERNST D. KRUPNIKOV Poste Restante Novosibirsk 90 630090, USSR EDITORIAL NOTE. For a comprehensive list of previous errata, see Math. Comp., v. 39, 1982, pp. 747-757 and the sources noted there. Except for the p. 681: 6.551.1 entry given herein, all errata als pertain to the 4th edition of 1964. 602.-MILTON ABRAMOWITZ & IRENE A. STEGUN, Editors, Handbook of Mathemati- cal Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Appl. Math. Series, No. 55, U. S. Government Printing Office. Washington, D.C., 1964, and all known reprints. On page 777, in formula 22.4.6 the value of Pn(O) should read 2()'(2m) for n = 2m. I. M. LONGMAN Department of Geophysics and Planetary Sciences Tel-Aviv University Ramat Aviv, Israel