gr7_errata

April 11, 2008 Erratafor 7thedition of G&R Page1 of 7 Errata for Tablesof Integrals,Series, and Products (7thedition) by I. S. Gradshteynand M. Ryzhik edited by Alan Jeffreyand Daniel Zwillinger, Academic Press, Orlando, Florida, 2007 ISBN 0-12-373637-4 http://www.mathtable.com/gr 4/11/2008 UPDATED: April 11, 2008 NOTES: 1Due to our procedures for verifying errata, the date that an entry is updated may be significantly later than the datethat the errata wasbrought to our attention. 2Thedate that an update to these errata pages is made is shownin the margin. 3Sometimesmanycontributorsbring the same errata to our attention. 4Thelatest errata is availablefrom http://www.mathtable.com/zwillinger/errata/ . ERRATA 1Acknowledgements ,pagesxxiii–xxv , The followingnames should be added 4/11/2008 •Dr.Ir.Luk R. Arnaut •Dr.Peter Arnold •Dr.Henry Corback •Ir.Enno Diekema •Dr.Grant Erdmann •Dr.Joel T.Johnson •Dr.JavierNavarroLaboulais•Dr.YefimLeifman •Dr.Angelo Melino •Dr.Haixing Miao •Dr.Ali Rushdi •Dr.SherwoodSamn •Dr.Detmar Welz 2Summation0.245.3 ,page 13, presently has the value 4/11/20081 e= 0.36787 . . . whichis incorrect. It should havebeen 1 2e= 0.1839397 . . . (Thanksto an anonymousreviewerfor correcting this error.) 3Formula0.434 ,page 23. The formula needs the followingconstraint: 4/11/2008 n < p (Thanksto Angelo Melino for correcting this error.) April 11, 2008 Erratafor 7thedition of G&R Page1 of 7 April 11, 2008 Erratafor 7thedition of G&R Page2 of 7 4Integral2.33 4 ,page 108, presently has an evaluationthat begins 4/11/2008 ±xm+1−n na whichis incorrect. It should havebeen ±xm+1−n nae±axn (Thanksto Henry Corback for correcting this error.) 5Integral2.33.16 ,page 109, presently has the integrand 4/11/2008 e−βxn whichis incorrect. It should havebeen e−βx2 (Thanksto VictorH. Moll for correcting this error.) 6Integral2.536 1 ,page 164, presently has an evaluationthat begins 4/11/20081 p+ 1. . . whichis incorrect. It should havebeen 1 p+a. . . (Thanksto SherwoodSamn and Grant Erdmann for correcting this error.) 7Integral3.1125 ,page253, presentlyhasanevaluationthatismissingaminussign(“ −”)infrontoftheentire 4/11/2008 expression. (Thanksto Haixing Miao for correcting this error.) 8Integrals in 3.323 ,page 337 . The following new integral should be added (which is a generalization of 4/11/2008 3.323.3): 3.323.4Z∞ 0exp¡ −β2x4±2γ2x2¢ dx=π 4γ βexpµγ4 2β2¶· I−1/4µγ4 2β2¶ ±I1/4µγ4 2β2¶¸ (Thanksto Joel T.Johnson for suggesting this addition.) 9Integral3.457 2 ,page 364, the evaluationis presently 4/11/2008 2 (2n+ 1)an+1/2· ln(4a)−3C−2ψ(2n)−ψ(n)¸ whichis incorrect, it should havebeen 2 (2n+ 1)an+1/2· ln(4a)−C−2ψ(2n) +ψ(n)¸ (Thanksto VictorH. Moll for correcting this error.) 10Integral3.471.1 ,page 367, is missing the constraint “ β >0”. 4/11/2008(Thanksto VictorH. Moll for correcting this error.) April 11, 2008 Erratafor 7thedition of G&R Page2 of 7 April 11, 2008 Erratafor 7thedition of G&R Page3 of 7 11Integral3.624 3 ,page 396, has an extrafactorialin the result. The result is presently 4/11/2008 π(2n)!! 22n+1(n!)2 whichis incorrect; it should havebeen π(2n)! 22n+1(n!)2 (Thanksto VictorH. Moll for correcting this error.) 12Integral 3.652 3 ,page 403, in two places the angle xis in an incorrect position. That is “ cotµ+1 2x” should be 4/11/2008 “cotµ+1 2x”and “ tanµ−1 2x” should be “ tanµ−1 2x”. (Thanksto VictorH. Moll for correcting this error.) 13Integral3.6543 ,page404, hastheangle xisanincorrectposition. Thatis“ tan±(µ−1)x”shouldbe“ tan±(µ−1)x”. 4/11/2008(Thanksto VictorH. Moll for correcting this error.) 14Integral3.691 3 ,page 415, the integrandnowends with an “ x”when is should end with a “ dx”. 4/11/2008(Thanksto VictorH. Moll for correcting this error.) 15Integral3.725.2 ,page 425, the integralshould be a principal valueintegral. 4/11/2008(Thanksto VictorH. Moll for correcting this error.) 16Integral3.852 5 ,page 476, the integrandis presently 4/11/2008¡ sin2x−x2cosx2¢1 x4 whichis incorrect; it should havebeen ¡ sinx2−x2cosx2¢1 x4 (Thanksto VictorH. Moll for correcting this error.) 17Integral3.852 6 ,page 476, the integrandis presently 4/11/2008 µ cos2x−1 1 +x2¶1 x whichis incorrect; it should havebeen µ cosx2−1 1 +x2¶1 x (Thanksto VictorH. Moll for correcting this error.) 18Integrals3.462 20–3.462 24 ,pages 476–477 , the integrandsall presently havethe term 4/11/2008 p x+b2 and,in everycase, it should be p x2+b2 (Thanksto VictorH. Moll for correcting this error.) April 11, 2008 Erratafor 7thedition of G&R Page3 of 7 April 11, 2008 Erratafor 7thedition of G&R Page4 of 7 19Integral3.931 4 ,page 495, the first integrandpresently has the limits 4/11/2008Zπ/2 0 whichis incorrect; it should havebeen Zπ 0 (Thanksto VictorH. Moll for correcting this error.) 20Integrals 3.948.5 and 3.948.6 ,page 500, each integrand contains “ e−βx” which is incorrect; they should have 4/11/2008 been“ e−px”. (Thanksto VictorH. Moll for correcting this error.) 21Integral 3.982.3 ,page 510, the denominator of the integrand is now “ sinh2hx” which is incorrect; it should 4/11/2008havebeen “ sinh2x’. (Thanksto VictorH. Moll for correcting this error.) 22Integral3.987.1 ,page 512, the result contains an ” α” which should be an “ a”. 4/11/2008(Thanksto VictorH. Moll for correcting this error.) 23Integral4.134.1 ,page524. Thisintegralisincorrectandshouldbedeleted. (Notethattheintegrandandlimits 4/11/2008 areidentical to integral4.134.2) (Thanksto VictorH. Moll for identifying this error.) 24Integrals4.135.1–2 ,page 524. Each of these integralscontains the term 4/11/2008 . . .µ −βγ2 α2+β2¶ . . . whichis incorrect. This term should havebeen . . .µβγ2 α2+β2¶ . . . (Thanksto Joel T.Johnson for correcting these errors.) 25Integral4.272.8 ,page 552. The evaluationof this integralis incorrect. The correct evaluationis: 4/11/2008 (n−1)!∞X k=0(−1)k (ν+k)n (Thanksto VictorH. Moll for correcting this error.) 26Integral4.272.18 ,page 552. The evaluationof this integralis incorrect. The correct evaluationis: 4/11/2008 Γµ 3−1 n¶³ p−3+1 n−q−3+1 n´ (Thanksto VictorH. Moll for correcting this error.) April 11, 2008 Erratafor 7thedition of G&R Page4 of 7 April 11, 2008 Erratafor 7thedition of G&R Page5 of 7 27Integral4.397.8 ,page 589, the integrandis presently 4/11/2008 ln¡ 1−2acosx+a2¢ sinnxsinx whichis incorrect, it should havebeen ln¡ 1−2acosx+a2¢ cosnxcosx (Thanksto VictorH. Moll for correcting this error.) 28Integral4.428.1 ,page 595. The evaluationof this integralis incorrect. The correct evaluationis: 4/11/2008 =πà a+blog 2 +mX n=1(−1)n(b−2an) n! where m≤b 2a< m + 1andm= 0,1,2,3, . . . (Thanksto YefimLeifman for correcting this error.) 29Integral4.537.3 ,page 604, the integrandis presently 4/11/2008 arctan³ tanλp 1−k2x2´ whichis incorrect, it should havebeen arctan³ tanλp 1−k2x2´ (Thanksto VictorH. Moll for correcting this error.) 30Integralsin 5.112 ,page 620. The followingnewintegralshould be added: 4/11/2008 5.112.14ZK(k) k4dk=−1 9k3h¡ 1 + 4 k2¢ E(k) + 2k/prime2K(k)i (Thanksto Detmar Welzfor suggesting this addition.) 31Integral5.52.2 ,page 629, presently has 4/11/2008 Z x−pZp+1(x)dx=−x−pZp(x) whichis incorrect. It should havebeen Z x−p+1Zp(x)dx=−x−p+1Zp−1(x) (Thanksto Detmar Welzfor correcting this error.) 32Integral6.153 ,page 633, is incorrect; it should be replaced with 4/11/2008 Z1 0xE(ax) (1−a2x2)√ 1−x2dx=π 2√ 1−a2 validwhen a2<1. (Thanksto Detmar Welzfor correcting this error.) April 11, 2008 Erratafor 7thedition of G&R Page5 of 7 April 11, 2008 Erratafor 7thedition of G&R Page6 of 7 33Integral6.155–6.157 ,page 633. The followingnewintegralsshould be added: 4/11/2008 6.155Z1 0xK(ax)√ 1−x2dx=π 2aarcsin( a) [ a2≤1] 6.156Z1 0xE(ax)√ 1−x2dx=π 4a³ arcsin( a) +ap 1−a2´ [a2≤1] 6.157Z1 0·µ 1−k2 2¶ K(k)−E(k)¸2µ 1−k2 2¶dk k7=1 48 (Thanksto Detmar Welzfor suggesting these additions.) 34Integral6.592.1 ,page 690, the integrationrange is presentlyZ∞ 0whichis incorrect. It should havebeenZ1 0 4/11/2008 35Integral6.625.10 ,page 704, the integrandis presently 4/11/2008 x−µ−1 2(x−1)µ−1e−αxKν(αx) whichis incorrect. It should havebeen x−µ−1 2(x−1)µ−1e−αxKν(αx) (Thanksto Theodoros Theodoulidis for this correction.) 36Formula8.241.1 ,page 887, presently has the term 4/11/2008 ···=xln ln1 x−. . . whichis incorrect and should be removed. (Thanksto Frank O’Brien for this correction.) 37Summations8.514.8–9 ,page 935. The followingnewsummations should be added: 4/11/2008 8.514.8∞X k=1(−1)k 2kJ2k(z) =−π 8N0(z) +1 4³ logz 2+C´ J0(z) 8.514.9∞X k=1(−1)k(2k+ 1) (2k+ 1)2−1J2k+1(z) =−π 8N1(z) +1 4³ logz 2+C−1´ J1(z)−1 4zJ0(z) (Thanksto Detmar Welzfor suggesting these additions.) 38Formula8.936.1 page 992, presently contains the term 4/11/2008 . . .½1 4³ t2−1´¾1 4−λ 2 . . . whichis incorrect. It should havebeen . . .½1 4³ 1−t2´¾1 4−λ 2 . . . (Thanksto Enno Diekemafor this correction.) 39Formula 8.982.2 ,page 1004 presently has an “ x” on the left hand side of the equation, which is incorrect; it 4/11/2008 shouldbe a “ z”. (Thanksto Luk Arnaut for this correction.) April 11, 2008 Erratafor 7thedition of G&R Page6 of 7 April 11, 2008 Erratafor 7thedition of G&R Page7 of 7 40Formula 9.303 ,page 1033 is presently incorrect. The first product sign in the numerator should have a prime 4/11/2008 onit, similar to the prime in 9.304. (Thanksto Peter Arnold for this correction.) 41Formula13.215.6.(ii) ,page 1073 . The formula is presently 4/11/2008 aiiaij>|aij|2fori/negationslash=j whichis incorrect. It should havebeen aiiajj>|aij|2fori/negationslash=j (Thanksto Ali Rushdi for this correction.) 42Statement14.12.1 ,page1075 . The statement nowbegins 4/11/2008 If anytwo adjacent rows(or columns) ... Whilethis is correct, it wouldbe better to say If anytworows(or columns) ... (Thanksto Ali Rushdi for this correction.) Datesof updates and errata numbers modified at those dates 4/11/2008 1–42 April 11, 2008 Erratafor 7thedition of G&R Page7 of 7