April 23, 2021 Errata for 8th edition of G&R Page 1 of 33 Errata for Tables of Integrals, Series, and Products (8th edition) by I. S. Gradshteyn and M. Ryzhik edited by Daniel Zwillinger and Victor Moll Academic Press, 2015 ISBN 0-12-384933-5 THIS UPDATE: April 23, 2021 NOTES • The home page for this book is http://www.mathtable.com/gr • The latest errata is available from http://www.mathtable.com/errata/ • The author can be reached at [email protected] • This edition of the errata includes all the corrections in the paper: Dirk Veestraeten, Some remarks, generalizations and misprints in the integrals in Gradshteyn and Ryzhik, SCIENTIA, Series A: Math- ematical Sciences, Vol. 26 (2015), pages 115–131. • This document contains new material following by corrections to the 8th editionl (starting on page 14). • The updates since the last set of errata (March 2020) are shown with the date in the margin, as this line has. 2021 NEW MATERIAL (TO ADD TO THE 8th EDITION) 1. On page xxxviii, add the following entry before “Weber function” 2021 E p(z) Exponential Integral 8.27 2. On page 220, add the following integral 2021 ¢ 2 x cos(xb) 2 2.641.13 e−cx dx a2 + x2 p π 2 aπ 2 2ac − b 2ac + b = p e−b =(4c) + ea c −e−ab − eab + erf p + eab erf p 2 c 4 2 c 2 c DO 3. On page 247, add section 2.9 Other Elementary Functions 4. On page 247, add section 2.91 Minimum & Maximum ¢ ¢ b v n−2 2.91.1 f(min xi; max xi) dx = n(n − 1) dv f(u; v)(v − u) du MAR2007 [a;b]n a a April 23, 2021 Errata for 8th edition of G&R Page 1 of 33 April 23, 2021 Errata for 8th edition of G&R Page 2 of 33 2.91.2 f(x; min xi; max xi) dx [a;b]n n ¢ ¢ X b v Y = dv du f (x; u; v j xj = u; xk = v) dxi n−2 j;k=1 a a [u;v] i2[n]nfj;kg j6=k MAR2007 5. Add section 2.92 Floor Function The floor of a number is the largest integer that is less than or equal to the number. For example b2:345c = 2 and b5c = 5. ¢ ¢ n 1 1 X nf(k) 2.92.1 ··· f (bx1 + ··· + xnc) dx1 ··· dxn = GR1994, #6.65, p 316, 557 0 0 k n! | {z } k=0 n n where the k are Eulerian numbers 6. Add section 2.93 Fractional Part of Numbers The fractional part of a number is fxg = x − bxc. For example f2:345g = 0:345 and f5g = 0. ¢ a+n n 2.93.1 fxg dx = [a > 0; n = 1; 2; 3;::: ] FUR2013, 2.42 2 ¢a 1 1 2.93.2 fkxg dx = [k = 1; 2; 3;::: ] FUR2013, 2.28 2 ¢0 1 1 2.93.3 fnxgk dx = [k > −1; n = 1; 2; 3;::: ] FUR2013, 2.44 k + 1 ¢0 1 (k!)2 2.93.4 (x − x2)k fnxg dx = [k = 0; 1; 2; : : : ; n = 1; 2; 3;::: ] 0 2(2k + 1)! ¢ FUR2013, 2.48 1 fxg 2.93.5 dx = 1 − C WOFP x2 ¢1 1 fxg 1 ζ(k) 2.93.6 dx = − [k = 2; 3; 4;::: ] FUR2013, 2.9 xk+1 k − 1 k ¢1 1 fxg − 1 p 2.93.7 2 dx = −1 + log( 2π) x ¢1 1 1 2.93.8 faxg − 1 fbxg − 1 dx = [Re a > 0; Re b > 0] WEBMHB 2 2 12ab ¢0 1 1 2.93.9 dx = 1 − C WOFP 0 x ¢ ( 1 n q o q(1 − C − log q) [0 < q ≤ 1] 2.93.10 dx = 1 1 bqc(fq}−1) 0 x q 1 + 2 + ··· + 1+bqc − C − log q + q(1+bqc) [q > 1] FUR2013, 2.5b April 23, 2021 Errata for 8th edition of G&R Page 2 of 33 th April 23, 2021¢ Errata for 8 edition of G&R Page 3 of 33 1 1 1 ζ(m + 1) 2.93.11 xm dx = − [m > 0] FUR2013, 2.20 x m m + 1 ¢0 1 x 1 2.93.12 dx = C FUR2013, 2.15 1 − x x ¢0 1 1 2 2.93.13 dx = log(2π) − 1 − C QIN2011 x ¢0 1 k 2 1 1 2.93.14 dx = k log(2π) − C + 1 + + ··· + + 2k log k − 2k − 2 log k! 0 x 2 k [k = 1; 2; 3;::: ] FUR2013, 2.6 ¢ 1 1 1 2.93.15 dx = 2C − 1 QIN2011 x 1 − x ¢0 ¢ 1=2 1 1 1 1 1 1 2.93.16 dx = dx = C − FUR2013, 2.10 0 x 1 − x 1=2 x 1 − x 2 ¢ 1 1 2 1 5 2.93.17 dx = − C − log(2π) FUR2013, 2.12 x 1 − x 2 ¢0 1 1 2 1 2 2.93.18 dx = 4 log(2π) − 4C − 5 QIN2011 x 1 − x ¢0 1 1 3 1 3 18ζ0(2) 2.93.19 dx = 6C + 2 − ζ(2) − 3 log(2π) − QIN2011 x 1 − x π2 ¢0 1 1 m ζ(2) + ζ(3) + ··· + ζ(m + 1) 2.93.20 xm dx = 1 − 0 x m + 1 [m = 1; 2; 3;::: ] FUR2013, 2.21 2.94 ¢ 1 1 k 2.94.1 p dx = − ζ(k)[k = 2; 3; 4;::: ] FUR2013, 2.7 k x k − 1 ¢0 1 k k 1 1 1 2.94.2 p dx = − kk ζ(k) − − · · · − [k = 2; 3; 4;::: ] k k k k 0 x k − 1 1 2 k FUR2013, 2.8 ¢ 1 1 1 ζ(k) 2.94.3 p dx = − [k = 2; 3; 4;::: ] FUR2013, 2.9 k k 0 k x k − 1 k 2.95 Combination of fractional part and other functions ¢ 1 1 1 2 2.95.1 (−1)b x c dx = 1 + log FUR2013, 2.13 x π ¢0 1 1 1 π2 1 2.95.2 x dx = − FUR2013, 2.14a x x 12 2 ¢0 1 m+1 m e 1 2.95.3 flog xg x dx = m+1 − 2 [m > −1] FUR2013, 2.16 0 (m + 1)(e − 1) (m + 1) April 23, 2021 Errata for 8th edition of G&R Page 3 of 33 April 23, 2021 Errata for 8th edition of G&R Page 4 of 33 2.96 Multiple integrals ¢ ¢ 1 1 x k 1 1 1 2.96.1 k dx dy = 1 + + ··· + − log k − C + 0 0 y 2 2 k 4 [k = 1; 2; 3;::: ] FUR2013, 2.28 ¢ ¢ 1 1 mx m n 3 2.96.2 dx dy = log + − C 0 0 ny 2n m 2 [m and n are integers with m ≤ n] FUR2013, 2.29 ¢ ¢ 1 1 xk 2k + 1 C 2.96.3 dx dy = − [k ≥ 0] FUR2013, 2.30 y (k + 1)2 k + 1 ¢0 ¢0 1 1 x y k ζ(2) + ζ(3) + ··· + ζ(k + 1) 2.96.4 dx dy = 1 − 0 0 y x 2(k + 1) [k = 1; 2; 3;::: ] FUR2013, 2.33 ¢ ¢ 1 1 x yk 1 ζ(2) + ζ(3) + ··· + ζ(k + 1) 2.96.5 p dx dy = − 0 0 y x k − p + 1 (k + 2 − p)(k + 1) [k is an integer, p is real, k − p > −1] FUR2013, 2.34 ¢ ¢ 1 1 x n y o π2 2.96.6 dx dy = 1 − FUR2013,2.36 y x 12 ¢0 ¢0 1 1 x2 log(2π) 1 C 2.96.7 dx dy = − − FUR2013, 2.31 y 2 3 2 ¢0 ¢0 1 1 x n y o 1 1 1 ζ(n + 2) ζ(m + 2) 2.96.8 xmyn dx dy = + − − 0 0 y x m + n + 1 n + 1 m + 1 n + 2 m + 2 [m > −1; n > −1] FUR2013, 2.37 ¢ ¢ 1 1 n x n y o 1 ζ(n + 1) 2.96.9 (xy) dx dy = 2 − 0 0 y x (n + 1) (n + 1)(n + 2) [n > −1] FUR2013, 2.38 ¢ ¢ 1 1 xm n y om ζ(2) + ζ(3) + ··· + ζ(m + 1) 2.96.10 dx dy = 1 − 0 0 y x m + 1 [m = 1; 2; 3;::: ] FUR2013, 2.40 ¢ ¢ 1 1 2x 2y 49 2π2 2.96.11 dx dy = − − 2 log 2 FUR2013, 2.39 y x 6 3 ¢0 ¢0 ¢ ¢ 1 1 x − y 1 1 x + y 1 2.96.12 dx dy = dx dy = FUR2013, 2.51 x + y x − y 2 ¢0 ¢0 0 0 1 1 k 1 1 1 2.96.13 dx dy = (1 − C)2 [k > 0] FUR2013, 2.52 x − y x y 2 ¢0 ¢0 1 1 1 ζ(2) π2 2.96.14 x dx dy = 1 − = 1 − FUR2013, 2.23 0 0 1 − xy 2 12 ¢ ¢ ( 2 1 1 1 m 2 log 2 − π m = 1 2.96.15 dx dy = 12 FUR2013, 2.24 x + y 5 π2 0 0 2 − log 2 − C − 12 m = 2 ¤ ( 2 1 m 2 log 2 − π m = 1 2.96.16 dx dy = 12 QIN2011, 3.1 x + y 3 π2 0≤x;y≤1 2 − 12 − log 2 − C m = 2 April 23, 2021 Errata for 8th edition of G&R Page 4 of 33 April 23, 2021 Errata for 8th edition of G&R Page 5 of 33 ¢ ¢ ¢ ( 1 m 9 log 3 − 13 − 19 log 2 − ζ(3) m = 1 2.96.17 dx dy dz = 2 24 4 3 x + y + z 53 ζ(3) π2 0≤x;y;z≤1 24 + 4 log 2 − 3 log 3 − 3 − 12 m = 2 ¢ ¢ QIN2011, 3.2 a1 an 1 2.96.18 ··· fk(x1 + x2 + ··· + xn)g dxn ··· dx1 = a1a2 ··· an FUR2013, 2.42b 0 0 2 7.