A1 for 1980 GR

Table Errata Source: Mathematics of Computation, Vol. 39, No. 160 (Oct., 1982), pp. 747-757 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2007357 Accessed: 19/04/2010 16:51 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 588. -JOHN BRILLHART, D. H. LEHMER & J. L. SELFRIDGE, "New primality criteria and factorizations of 2 +? 1," Math. Comp., v. 29, 1975, pp. 620-647. P. 639, equation 36: The colon should be placed directly after the factor 257. P. 641, equation 88: The ending for the final factor should read ... 69361. P. 643, line -9: Insert 225 between 222 and 226. (Submitted to me by Oswald Wyler.) P. 643, line - 1: Insert 205 between 201 and 207. P. 644, line 13: Read "nine" for "eight", and insert 233 between 197 and 239. JOHN BRILLHART Department of Mathematics University of Arizona Tucson, Arizona 85721 589.-I. S. GRADSHTEYN & I. M. RYZHIK, Table of Integrals, Series, and Products, Corrected and enlarged edition, Academic Press, New York, First printing, 1980. FOREWORD This errata list, relating to the recently published "Table of Integrals, Series, and Products", corrected and enlarged edition, by I. S. Gradshteyn and I. M. Ryzhik, Academic Press, New York, 1980, is an abridged version of a more extended one which we have edited in March, 1981. In the interest of brevity, only the more significant items are included in this abridged edition. For instance, conditions like a > 0, n = 1, 2,. .., given by Gradshteyn and Ryzhik, are often only sufficient for the validity of a formula. Frequently the domain of validity can be extended into the complex plane by analytic continuation. The lack of precision in the formulation of the conditions in such cases is not considered as an erratum. These and similar items are omitted. A number of valid errata which refer to previous editions of Gradshteyn and Ryzhik and which remain in the corrected and enlarged edition have been previously reported in Mathematics of Computation (see the references below) and are therefore also omitted. We have corrected a number of invalid errata reported in Mathematics of Compu- tation. These corrections are reported herein. We have incorporated many corrections that are listed on pp. 1101-1108 of the Russian fifth edition, published in Moscow in 1971. New errata, recently submitted to us by H. E. Fettis, E. A. Kuraev, and Y. L. Luke have also been incorporated. We acknowledge with thanks their contributions along with comments and suggestions on our work. A more extensive list of errata for the corrected and enlarged edition and a corresponding list for the fourth edition of Gradshteyn and Ryzhik are available from the authors on request. 747 748 TABLE ERRATA Finally, listed below are the Mathematical Table Errata and Corrigenda notices relating to this work, together with the volume of Mathematics of Computation in which they can be found. H. VAN HAERINGEN L. P. KOK Department of Mathematics Delft University of Technology Delft, The Netherlands Institute for Theoretical Physics P. 0. Box 800 Groningen, The Netherlands MTE #408: v. 21(1967), pp. 293-294. MTE #428: v. 22(1968), pp. 903-907. MTE #437: v. 23(1969), pp. 468-469. MTE #446: ibid., pp. 891-892. MTE #473: v. 25(1971), p. 200. MTE #486: v. 26(1972), p. 305. MTE #492: ibid., p. 599. MTE #503: v. 27(1973), pp. 451-452. MTE #528: v. 30(1976), p. 899. MTE #534: v. 31(1977), p. 614. MTE #550: v. 32(1978), p. 318. MTE #557: v. 33(1979), p. 430. MTE #564: ibid., p. 846. MTE #565: ibid., p. 1377. MTE #572: v. 35(1980), p. 1444. MTE #577: v. 36(1981), pp. 317-318. MTE #582: ibid., p. 320. Review, v. 20(1966), pp. 616-617. Review, v. 36(1981), pp. 310-314. Corrigendum, v. 26(1972), p. 601. Corrigenda, v. 33(1979), p. 433. p. xl: Left column, line 9 up Read j(x, /B), A(x, /3, a) instead of j(x, /3). p. 9: 0.243.2 Replacep + (k-l)q + 1 byp + (k1-)q + 1. p. 27: 1.331.1 For (n) read (n). 1.33 1.3 For () read (:). 1.331.4 For (2nk) read (nk). p. 38: 1.443.1 Read (x) instead of (X). p. 40: 1.445.2 Read 0 < x instead of 0 < x. p. 46: 1.518.1 Read 0 < x < 77 instead of x2 < 772. p. 46: 1.518.3 Read 0 < x < 77/2 instead of x2 < 772/4. p. 46: 1.521.1 Add [-?T/2 < x < 7T/2]. p. 46: 1.521.2 Add [0 < x < 7T]. p. 62: Line 6 Replace a/ by a/ . p. 94: 2.413.2 Second term on right-hand side: Read instead of TABLE ERRATA 749 p. 149: 2.557.2 Read a sin x + b cos x instead of a cos x + b sin x. p. 159: 2.583.11 2(2k 4- k 2-1) 2(2k 4- k 2-1) Read - 15k4 instead of + 15k4 15k4 15k4 p. 169: 2.586.8 Read 1 + k sin x instead of 1 + sin x. p. 285: 3.194.5 Read arg(l + upi) instead of arg(l - up). p. 288: 3.219 Read Re A> -1, Re v > -1 instead of Re A> 1, Re v > 1. p. 290: 3.228.3 Add [A + v #L 1, A #L 1, 2,...] to the second right-hand side. p. 292: 3.241.2 Read Re v > instead of Re v . p. 301: 3.271.5 Read 0 <p2 < I instead of p2 < p. 304: 3.341.5 Read 3.265 instead of 3.266. p. 306: 3.317.1 Read 3.233 instead of 3.2332. p. 307: 3.323.1 is incorrect since the sum is divergent. According to 3.322.1 the integral is equal to IIT 1/2eq2/4[I- D(i + 2q)]. When q = -2 this expression reduces to I T'/2e. For q #? -2 one obtains, by using 8.254 and (2k - 1)!!= 7T-1/22kr(k + 2), k = 0,1,2,..., an asymptotic expression which is in agreement with the right side of 3.323.1. p. 308: 3.333.2 Add [A # 1]. For I = 1 the value of the integral is ln 2. p. 309: 3.337.2 Read HV(') instead of H^. p. 309: 3.337.3 Read HV(2) instead of HV2. p. 311: 3.352.5 Add: If a > u, the integral must be interpreted in the Cauchy principal value sense. See the discussion given in this report for p. 523, 4.212. Similar remarks pertain to all integrals on pp. 311-315 where the symbol Ei(x) is used. p. 311: 3.353.3 should read ePxdx peaPEi(-ap) + [p > 0, a > 0]. 0(a+ X)2a p. 319: 3.383.6 Read Rej 2 0 instead of Re A > 0. p. 319: 3.383.8 Delete the first three lines of the six lines on the right-hand side. p. 321: 3.387.1 Read Rev> 0 instead of Re z 0. p. 324: 3.394 Replace 1 - 2iA by -2iA and Re v > 0 by Re v < O. p. 325: 3.411.6 Read x'- i instead of e'- i. p. 325: 3.411.9 Read 4.231 3. instead of 4.231 2. p. 326: 3.411.19 and 3.411.20 Delete[ ] p. 326: 3.411.22 For the right-hand side read 00 k= 1 p. 326: 3.411.25 Read 4.231 4. instead of 4.231 3. p. 328: 3.417.1 Read 4.23 1 8. instead of 4.23 1 6. 750 TABLE ERRATA p. 328: 3.417.2 Read 4.231 10. instead of 4.231 8. p. 329: 3.418.1 This integral, p. 354, 3.531.1 and the negative of p. 533, 4.233.2 are the same. Note the difference in the numerical values. We presume that the latter is correct. p. 329: 3.419.4 Read [ T2 + (ln /)2]2 instead of ?T2 + (ln /)2. p. 329: 3.419.6 Read (11) instead of (7). p. 331: 3.423.5 Read 4.231 5. instead of 4.231 3. p. 331: 3.425.1 Read 4.231 7. instead of 4.231 5. p. 337: 3.458.2 Read v = 1,2, .. . instead of v-an integer. p. 337: 3.461.5 Read I arg A I < 7/2 instead of I arg A I < ?T/4. p. 342: 3.478.1 Replace l/lp I by I/p and add [p > 01. p. 343: 3.482.1 Read -7N,(fi) instead of +?TN(,B). p. 343: 3.483 The right-hand side should read 2exp(- 2ivi)K,(a) fora > 0, 2exp(2iv?Ti)K,(-a) for a < 0 [j Repv < 1]. p. 345: 3.514.1 Add [a > 0]. p. 345: 3.514.2 Read 0 < a j< b, O < t2 < STinstead of b >I a l, 0 < t < ?T. p. 345: 3.514.3 Read 0 < a2 < I instead of a2 < 1. p. 345: 3.514.4 Read 0 <1 b j< a instead of a >1 b I . p. 350: 3.524.9 is incorrect since the integral is divergent. Perhaps the integral should read J_ x_2m-_Ie_ ax J0 ch ax See A. Erdelyi et al., Higher Transcendental Functions, Vol. 1, p. 39, Eq. (25). p. 352: 3.527.3 Add [A #& 2]. Add = $2ln 2 if =2. p. 353: 3.527.6 Read Rej A> 1 instead of Re A > 0. p. 354: 3.53 1.1 Add: - 31/2 [3 ln -LQr7/3)] 31/2 Cl2(0/3) where L(x) is given on p. 933, Section 8.260, and CI2(x) is Clausen's integral. See also our comments on p. 329, 3.418.1. p. 354: 3.531.2 For sin t cos t read sin 2t. p. 354: 3.531.5, 3.531.6, 3.531.7. Note that these integrals are related. In 3.531.6, use p. 1075, 9.550, replace A by j + 1 and t by ?T- t to get 3.531.7. In 3.531.6 if t 0 O the integral follows from p. 352, 3.527.1. Also if t = ?T then 3.531.6 follows from p. 352, 3.527.3. In 3.531.6 if t = ?T and A = 2 then the integral follows from 3.531.2. In 3.531.7, let A = 2m and t = T- 2a?T. Then 3.531.5 emerges. It is sufficient to cite conditions for 3.531.7. Thus 0 < t < 2iT, Re(A) > -1 unless t = ?Tin which case Re(A) > 1. Values of the right-hand side for t = 0 or t = 2 ? and t = ?Tare easily found by taking limits. For t = Oort = 2S7, Re(y) > 0. TABLE ERRATA 751 p. 355: 3.533.2 Read 7T t t 2 cosec t instead of 2 cosec t. a2 a2 Add [a > 0]. p. 355: 3.533.4 Read [0 < a < 1,a a 4] instead of [0 < a < T]. Read sin 2ka7r instead of cos 2ka r. Add = 2(2m + 1)(22m-1 - l)?T2m I B2m I [a ] p. 356: 3.541.2 Read EH I 16(14)a instead of EH 1 16. p. 356: 3.541.4 Add [0 < a < 2]. p. 356: Note that 3.541.10 and 3.541.4 are similar. p. 357: 3.544 Add [u > 01. p. 361: 3.552.2 and 3.552.4 Add [a > 0, m = 1, 2,...]. p. 361: 3.552.3 Add [A 7P 1]. Add = ln2 if= 1. p. 362: 3.555.1 Read [0 <21 a <p] instead of [2a <pl. p. 364: 3.557.6 Read I a j r instead of a?T. p. 367: 3.613.4 First line: Add [n 2 1]. Second line: Add [n > 1]. Add =- 7 2 [n = O, a2 < 1] - a(a2-l) [n = 0, a > 1]. a(a 2- 1) p. 371: 3.624.6 The right member should read = Ta - sin 7Ta[2a13(a) - 1], a >0. Cf. [MTE 582] with x replaced by 4 on the right-hand side. p.391: 3.687.1 Add[Re(p.+v)<2]. p. 392: 3.687.2 Add [Re(A + v) < 4]. p. 395: 3.691.8 and 3.691.9 are incorrect. Note that these integrals follow from 3.691.4, 5 and 3.691 6, 7, respectively. p. 405: 3.718.7 Read T(1 - e-a) instead of (1 - e-a). p. 414: 3.739.2 Add [a = 01 to the second right-hand side. p. 415: 3.742.2 Read 13(a + b) instead of f(a + P3). p. 415: 3.742.6 Read b > a > 0 instead of 0 < b < a. p. 416: 3.745.1 and 3.745.2 are incorrect. The integrals are divergent. p. 424: 3.768.3 The function CQ(a) on the right-hand side is not defined. Note that 3.768.3 is a special case of 3.768.5. p. 424: 3.768.5 An expression for the integral which is valid for Re v > -1 is: auu+vB(A, P + 1) 2F3 ( 2 A 2 P2 2 '2 ' 2 A+ v+ 2 3 2U2 2 ' 2 ' 4 A 752 TABLE ERRATA p. 424: 3.768.6 The expression is correct, but awkward to use. Add = u+^lB(,, ^)2F3( 2 ,U+ V ? + V + I -a2U2 2' 2 ' 4J} p. 425: 3.768.11 Read Rev > -1, v #?0 instead of Rev >0. Read ETI 68 instead of ETI 58. Note that 3.768.11 and 12 are essentially the same as 3.768.5 and 6, respectively. p. 425: 3.768.13 Read sin(2ax) instead of sin(ax). p. 425: 3.768.14 Read cos(2ax) instead of cos(ax). p. 427: 3.771.5 Except for the condition on v which should read Re v < 0, the entry is correct but awkward to use. A more convenient expression is 0? X(X2 + .B2)-1/12- snaxu + fi21/>!sin axdx ITI 2# au ,'_I(afi) IF( I +I)( 2P)K la [a > 0, Re(/A) > 0, Re(j ) > 0]. p. 436: 3.792.10 The integral should read 00o 1 dx Jo 1- 2acosbx + a2 /32 + X2 p. 437: 3.793.1 Read -2,ra' instead of 2,Tan. p. 445: 3.819.4 Read Jo instead of Jo. p. 448: 3.824.5. The given result is incorrect. Read - Te-a(2m + 1)! F(-m,l e2a" 22m+lm!(m + 1)! 2 1m+ 2 [I arg a j< ?T/2], m = 0, 1, 2,.... If a = 0, value of the integral is ?T(2m)!/22m+ i(rm!)2. Note: In many of the entries on pp. 447-449 and elsewhere, the parameter a or its analog can be complex provided it is restricted as above. Also a can be zero. p. 454: 3.832.2 Add [m = 0, 1, ... ]. Add for m = - 1, see 3.723 2. p. 454: 3.832.6 is incorrect. The right-hand side should read (i )T ear( - e-2a)2m -(1+ e-2a)] [a > 0, m = 0, 1,...1. p. 455: 3.832.18 is incorrect. The right-hand side should read (f { r(e ar(1 - e2a)2ma - I] - ea} [a 2 0, m =0, 1,...]. TABLE ERRATA 753 p. 455: 3.832.26 is incorrect. The right-hand side should read IT cha[(1 + e-2a)m1 - I] [a0, m =0,1,...]. p. 456: 3.832.31 is incorrect. The right-hand side should read 27ea[(1 + e2a)r-(1- e-2a)] [a >0, m = 0,1,...]. p. 458: 3.836.5 Replace n + an + 2k by n + an - 2k. Also replace LO V 340(14) by ET I 20(11) and delete the remark in square brackets. See Math. Comp., v. 36, 1981, pp. 312, 313. On the latter page, line 7, for 3.836.5 read 3.836.4. p. 458: 3.836.6 Read n > 2 instead of n = 2. p. 480: 3.898.3 Replace I V by 4v. p. 481: 3.911.3 Read T th a?T instead of - 4 th(a?T). p. 505: 3.983.3 Read cth a?T instead of ch a?T. Delete BI((267))(4), Note that the integral is of the Cauchy principal-value type. p. 506: 3.984.1 and 3.984.2 are incorrect: The integrals are divergent. p. 507: 3.985.3 Read [Refi > 0, n = 0, 1,. . . ,all real a] instead of [a > 0]. p. 523, 4.212; pp. 524, 525. There is considerable confusion in this section. Equation 4.212.1 must be written as IV dx ___ e-aPV 00tt P.V. = P.V. _dy =-e-ap.V. e t-Iet dt, a>0, a + Inx ~a -y -a where P.V. stands for Cauchy principal value. Thus -Ei(x) =P.V. tf Ve-tdt =lim( tfvle-tdt + tf le-tdt) as 0 0, - > 0, x > 0. See p. 925, 8.211.2, pp. 310-315 and EH, v. 2, p. 143. We have the equivalent notations Ei (x) = Ei(x) = E*(x). Thus p. 311, 3.352.6 should read P.V. a xdx = e-,aEi(aA). Note that -Ei(-a) J l7t-le-t dt and that the expression is valid for I arg a I < ?T. See p. 310, 3.351.5 and p. 925, 8.211.1. In the EH reference cited above, El(a) = -Ei(-a). The integrals in 4.212.3,.5,.8 are all divergent. They can be considered valid if we interpret jxf(y) dy d n(-I x~*f(y) dy, << ,()0 JO (y n = (n l ) ! dt ) P.V. I y) e < t < x, f(t) =# O On pp. 524, 525, 925, 926, most integrals expressed in terms of Ei(a) or Ei(x) must be interpreted in the Cauchy principal-value sense. 754 TABLE ERRATA p. 527: 4.224.11 is incorrect for the cases a2 < 1 and a2 > 1. According to MTE 503, for a > 0 the common value of the two integrals can be written as x0 bk k ( I )n+l I(a) =4G + rln(a/2) +4~ 7T_n- where G is Catalan's constant and b = (1 - a)/(l + a). An alternative result valid for a 2 0 is I(a) 4G + ?Tln( + a - 4S*(b), S*(b)= 2 b - 2 ( 1j' b (1- a)/(l + a). In the editorial note to MTE 503, the first and second terms should read ITn + (1- a2)1/2}an { 2 } -2(arcsina)ln{1 +(1-a2)l} respectively. We also have 1(a) = 7Tln{ 1 + (1- a2)/2} 00 (-1)k+l a 2k- 1 +4 21 a)12 a k=- (2k 1)\ 1 + (1-a2) ) I(a) =T Irn 2 + a k1 (2 - )2 cos(2k -1)0, 0 =arccos(l/a),a2>1. p. 533: 4.233 Analytical expressions for the first four integrals are 4.233.1 2[2 ?T2 () 9 [3 (3)] 4.233.2 1[272 (+,1)] TABLE ERRATA 755 4.233.3 9[6 ( 3)] 4.233.4 6 [ 6 (3)] p. 537: 4.251.3 Read f3'(j) instead of 2A(y) p. 538: 4.254.4 Read A 'p) instead of 2A(p) p. 539: 4.261.1 Add [0 < t < 7T]. For t = 0 or t = g take the limit of the right-hand sides to get ?T2/6 and ?T 2/3, respectively. p. 540: 4.261.4 Add [a > 01. p. 540: 4.261.8 Read 1 [4V'3T __?T 3 [ 27 instead of 27 p. 549: 4.271.14 Replace J0' by Jf. On the right-hand side, replace cos t by cosec t, and j t j < 7 by 0 <jt j< T. p. 549: 4.271.16 Read 1n} 8( ) P )instead of q?(P) p. 550: 4.272.3 Replace Jo by Jol. On the right-hand side, for I' o read k= For 0 < Re v < 2 read Re v > 0. For j t j < ?T, read -?T < t < ?T. p. 566: 4.311.1 is incorrect: The integral is divergent. Perhaps Jo ln 1- x31 dx/x3 is intended. If so, the value is -?T(3)1/2/6. See p. 558, 4.293.7. p. 571: 4.325.1 The value of this integral is - '(ln 2)2. p. 577: 4.355.2 should read I = 41 I+ -exp ( I4 + ( 4,A 4A A PI7[1 p. 578: 4.358.1 The right-hand side should read a m {A-VIF(v, A)) [m = 0,91,92,.] [ReA > 0, Rev > 0]. p. 582: 4.376.8 Add [n = 1, 2, ... ]. For n = 0 the value of the integral is 1/a. p. 582: 4.376.9 Add [a > 0, n = 1, 2, ... ]. For n = 0 the integral is divergent. 756 TABLE ERRATA p. 605: 4.441.1 Read -In instead of t- P In. 22 p. 634: 5.53 Read q(/X) instead of P(/:x) on the left-hand side. p. 640: 6.214.2 Read 0 < p < 1 instead of p > 0. p. 641: 6.224.1 Second line: Read = - 1/,B instead of = 1. p. 643: 6.244.1 and 6.244.2 Delete + r/2 in the integrand. p. 658: 6.422.6 Read 4,gi instead of -4qTi. p. 660: 6.423.3 and 6.423.4 The right member should be multiplied by F(m + 1). p. 673: 6.522.7 Read b-2(l + 4a2b-2)-'/2 instead of b-2(l + 4a2b-2)1/2 p. 679: 6.539.1 Add J,(x) #0 for x E [a, b]. p. 679: 6.539.2 Add N,(x) 4-0 for x E [a, b]. p. 733: 6.672.8 Read 24b instead of 2ab. p. 769: 6.784.2 Multiply the right member by 2 Vv. Also a l/2-- should be lowered. p. 833: 7.343.2 First line: Delete or m = n = 0. Second line: Delete 4- 0. p. 837: 7.374.5 Add a2 4- 4. For a2 =- Isee 7.374.3. p. 838: 7.378 Read 22m instead of 22m. p. 843: 7.393.2 Read F(2n + v + 1) instead of F(2n + p 1). p. 849: 7.512.6 Read (1 - zb b)a instead of F(a, b; /3; zlb). p. 849: 7.512.9 Read (1 - z)-a instead of (1 - z)a. p. 904: Line 8 Read (1 -nx2) instead of (1 + nx2). Line 14 Read BY(1 10.04) instead of Fl II 97-106. p. 904: 8.110.2 Read 1-n sin2 (p instead of 1 + n sin2 (P. Read BY(1 10.04) instead of Fl 11 106. p. 905: 8.111.4 Read 1-n sin2 a instead of 1 + n sin2 a. Read 1 - nx2 instead of 1 + nx2. Read BY(1 10.04) instead of SI 13. p. 909: 8.130.8 should read: A single-valued function (which is not constant).... p. 925: 8.211.3 Read Ei(x) instead of Ei. pp. 925, 926: See the comments concerning pp. 523-525. p. 937: 8.332.4 Read ch 2 y7 instead of sh 2 yv. p. 945: Line 3 Read 9.237 instead of 9.238. p. 947: 8.371.1 The integral should have limits 0 and 1. p. 960: 8.442.1 Replace F(M + k + 1) by k!F( i + k + 1). p. 960: 8.444.2 Replace {**}by lk+ 2E-. Note that Ek-=1 for k =1 vanishes, according to the convention at page xliii. This corrects the corresponding entry in Math. Comp., v. 36, 1981, p. 314. TABLE ERRATA 757 p. 973: 8.512.2 Add[n= 1,2, ...I. For n - 0 see 8.512.1. p. 1030: 8.933.4 Read CQ- instead of C,-1. p. 1038: 8.974.4 Read L nm(y) instead of LO fj(x). p. 1042: Line 21 Read 7.725 6 instead of 7.726 6. p. 1066: 9.246.2 Read (p + l)(p + 2) ... instead of p(p + l)(p + 2) p. 1067: 9.254.2 Delete the minus sign in front of the right-hand side. p.1094: 11.118 Line 3: Read bj instead of hi.