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The following corrections and other changes have been made in the DLMF, and are pending for the Handbook of Mathematical Functions. The Editors thank the users who have contributed to the accuracy of the DLMF Project by submitting reports of possible errors. For confirmed errors, the Editors have made the corrections listed here.

Printable errata PDF.

Version 1.0.13 (September 16, 2016)

Other Changes

  • In applying changes in Version 1.0.12 to (13.9.16) an editing error was made; it has been corrected.

Version 1.0.12 (September 9, 2016)

Equations (25.11.6), (25.11.19), and (25.11.20)

Originally all six integrands in these equations were incorrect because their numerators contained the function B~2(x). The correct function is B~2(x)-B22. The new equations are:

25.11.6 ζ(s,a)=1as(12+as-1)-s(s+1)20B~2(x)-B2(x+a)s+2dx,
s1, s>-1, a>0.

Reported 2016-05-08 by Clemens Heuberger.

25.11.19 ζ(s,a)=-lnaas(12+as-1)-a1-s(s-1)2+s(s+1)20(B~2(x)-B2)ln(x+a)(x+a)s+2dx-(2s+1)20B~2(x)-B2(x+a)s+2dx,
s>-1, s1, a>0.

Reported 2016-06-27 by Gergő Nemes.

25.11.20 (-1)kζ(k)(s,a)=(lna)kas(12+as-1)+k!a1-sr=0k-1(lna)rr!(s-1)k-r+1-s(s+1)20(B~2(x)-B2)(ln(x+a))k(x+a)s+2dx+k(2s+1)20(B~2(x)-B2)(ln(x+a))k-1(x+a)s+2dx-k(k-1)20(B~2(x)-B2)(ln(x+a))k-2(x+a)s+2dx,
s>-1, s1, a>0.

Reported 2016-06-27 by Gergő Nemes.

Other Changes

  • The symbol is used for two purposes in the DLMF, in some cases for asymptotic equality and in other cases for asymptotic expansion, but links to the appropriate definitions were not provided. In this release changes have been made to provide these links.

  • A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added in §2.1(iii) below (2.1.16).

  • Because (2.11.4) is not an asymptotic expansion, the symbol that was used originally is incorrect and has been replaced with , together with a slight change of wording.

  • Originally (13.9.16) was expressed in term of asymptotic symbol . As a consequence of the use of the O order symbol on the right hand side, was replaced by =.

  • In (13.2.9) and (13.2.10) there were clarifications made in the conditions on the parameter a in U(a,b,z) of those equations.

  • Originally (14.15.23) used f(x) to represent both U(-c,x) and U¯(-c,x). This has been replaced by two equations giving explicit definitions for the two envelope functions. Some slight changes in wording were needed to make this clear to readers.

  • The title for §17.9 was changed from Transformations of Higher ϕrr Functions to Further Transformations of ϕrr+1 Functions.

  • A number of additions and changes have been made to the metadata in Chapter 25 Zeta and Related Functions to reflect new and changed references as well as to how some equations have been derived.

  • Bibliographic citations, clarifications, typographical corrections and added or modified sentences appear in §§18.15(i) and 18.16(ii).

Version 1.0.11 (June 8, 2016)

Figure 4.3.1

This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

See accompanying text

Reported 2015-11-12 by James W. Pitman.

Equation (9.7.17)

Originally the constraint, 23π|phz|<π, was written incorrectly as 23π|phz|π. Also, the equation was reformatted to display the constraints in the equation instead of in the text.

Reported 2014-11-05 by Gergő Nemes.

Equation (10.32.13)

Originally the constraint, |phz|<12π, was incorrectly written as, |phz|<π.

Reported 2015-05-20 by Richard Paris.

Equation (10.40.12)

Originally the third constraint π|phz|<32π was incorrectly written as π|phz|32π.

Reported 2014-11-05 by Gergő Nemes.

Equation (23.18.7)

23.18.7 s(d,c)=r=1c-1rc(drc-drc-12),

Originally the sum r=1c-1 was written with an additional condition on the summation, that (r,c)=1. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare.

Equations (28.28.21) and (28.28.22)

28.28.21 4π0π/2𝒞2+1(j)(2hR)cos((2+1)ϕ)ce2m+1(t,h2)dt=(-1)+mA2+12m+1(h2)Mc2m+1(j)(z,h),
28.28.22 4π0π/2𝒞2+1(j)(2hR)sin((2+1)ϕ)se2m+1(t,h2)dt=(-1)+mB2+12m+1(h2)Ms2m+1(j)(z,h),

Originally the prefactor 4π and upper limit of integration π/2 in these two equations were given incorrectly as 2π and π.

Reported 2015-05-20 by Ruslan Kabasayev

Other Changes

Version 1.0.10 (August 7, 2015)

Section 4.43

The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.

Let p(0) and q be real constants and

4.43.1 A =(-43p)1/2,
B =(43p)1/2.

The roots of

4.43.2 z3+pz+q=0


  1. (a)

    Asina, Asin(a+23π), and Asin(a+43π), with sin(3a)=4q/A3, when 4p3+27q20.

  2. (b)

    Acosha, Acosh(a+23πi), and Acosh(a+43πi), with cosh(3a)=-4q/A3, when p<0, q<0, and 4p3+27q2>0.

  3. (c)

    Bsinha, Bsinh(a+23πi), and Bsinh(a+43πi), with sinh(3a)=-4q/B3, when p>0.

Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

Reported 2014-10-31 by Masataka Urago.

Equation (9.10.18)

9.10.18 Ai(z)=3z5/4e-(2/3)z3/24π0t-3/4e-(2/3)t3/2Ai(t)z3/2+t3/2dt.

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

Equation (9.10.19)

9.10.19 Bi(x)=3x5/4e(2/3)x3/22π0t-3/4e-(2/3)t3/2Ai(t)x3/2-t3/2dt.

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

Equation (10.17.14)

10.17.14 |R±(ν,z)|2|a(ν)|𝒱z,±i(t-)exp(|ν2-14|𝒱z,±i(t-1))

Originally the factor 𝒱z,±i(t-1) in the argument to the exponential was written incorrectly as 𝒱z,±i(t-).

Reported 2014-09-27 by Gergő Nemes.

Equation (10.19.11)

10.19.11 Q3(a)=54928000a8-1 107676 93000a5+7912375a2

Originally the first term on the right-hand side of this equation was written incorrectly as -54928000a8.

Reported 2015-03-16 by Svante Janson.

Equation (13.2.7)

13.2.7 U(-m,b,z)=(-1)m(b)mM(-m,b,z)=(-1)ms=0m(ms)(b+s)m-s(-z)s

The equality U(-m,b,z)=(-1)m(b)mM(-m,b,z) has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation a=-n has been changed to a=-m.

Reported 2015-02-10 by Adri Olde Daalhuis.

Equation (13.2.8)

13.2.8 U(a,a+n+1,z)=(-1)n(1-a-n)nza+nM(-n,1-a-n,z)=z-as=0n(ns)(a)sz-s

The equality U(a,a+n+1,z)=(-1)n(1-a-n)nza+nM(-n,1-a-n,z) has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

Reported 2015-02-10 by Adri Olde Daalhuis.

Equation (13.2.10)

13.2.10 U(-m,n+1,z)=(-1)m(n+1)mM(-m,n+1,z)=(-1)ms=0m(ms)(n+s+1)m-s(-z)s

The equality U(-m,n+1,z)=(-1)m(n+1)mM(-m,n+1,z) has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation a=-m,m=0,1,2, has been introduced.

Reported 2015-02-10 by Adri Olde Daalhuis.

Equation (18.33.3)

18.33.3 ϕn*(z)=znϕn(z¯-1)¯=κn+=1nκ¯n,n-z

Originally this equation was written incorrectly as ϕn*(z)=κnzn+=1nκ¯n,n-zn-. Also, the equality ϕn*(z)=znϕn(z¯-1)¯ has been added.

Reported 2014-10-03 by Roderick Wong.

Equation (34.7.4)

34.7.4 (j13j23j33m13m23m33){j11j12j13j21j22j23j31j32j33}=mr1,mr2,r=1,2,3(j11j12j13m11m12m13)(j21j22j23m21m22m23)(j31j32j33m31m32m33)×(j11j21j31m11m21m31)(j12j22j32m12m22m32)

Originally the third 3j symbol in the summation was written incorrectly as (j31j32j33m13m23m33).

Reported 2015-01-19 by Yan-Rui Liu.

Other Changes

Version 1.0.9 (August 29, 2014)

Equation (9.6.26)

9.6.26 Bi(z)=31/6Γ(13)e-ζF11(-16;-13;2ζ)+37/627/3Γ(23)ζ4/3e-ζF11(76;73;2ζ)

Originally the second occurrence of the function F11 was given incorrectly as F11(76;73;ζ).

Reported 2014-05-21 by Hanyou Chu.

Equation (22.19.6)

22.19.6 x(t)=cn(t1+2η,k)

Originally the term 1+2η was given incorrectly as 1+η in this equation and in the line above. Additionally, for improved clarity, the modulus k=1/2+η-1 has been defined in the line above.

Reported 2014-05-02 by Svante Janson.

Paragraph 22.19(ii)

Two corrections have been made in this paragraph. First, the correct range of the initial displacement a is 1/β|a|<2/β. Previously it was 1/β|a|2/β. Second, the correct period of the oscillations is 2K(k)/η. Previously it was given incorrectly as 4K(k)/η.

Reported 2014-05-02 by Svante Janson.

Equation (34.3.7)

34.3.7 (j1j2j3j1-j1-m3m3)=(-1)j1-j2-m3((2j1)!(-j1+j2+j3)!(j1+j2+m3)!(j3-m3)!(j1+j2+j3+1)!(j1-j2+j3)!(j1+j2-j3)!(-j1+j2-m3)!(j3+m3)!)12

In the original equation the prefactor of the above 3j symbol read (-1)-j2+j3+m3. It is now replaced by its correct value (-1)j1-j2-m3.

Reported 2014-06-12 by James Zibin.

Other Changes

Version 1.0.8 (April 25, 2014)

Equation (22.19.2)

22.19.2 sin(12θ(t))=sin(12α)sn(t+K,sin(12α))

Originally the first argument to the function sn was given incorrectly as t. The correct argument is t+K.

Reported 2014-03-05 by Svante Janson.

Equation (22.19.3)

22.19.3 θ(t)=2am(tE/2,2/E)

Originally the first argument to the function am was given incorrectly as t. The correct argument is tE/2.

Reported 2014-03-05 by Svante Janson.

Other Changes

  • Minor additions have been made in §§9.6(iii), 22.19(i).

  • Equation (10.13.4) has been generalized to cover an additional case.

  • We avoid the troublesome symbols, often missing in installed fonts, previously used for exponential e, imaginary i and differential d.

Version 1.0.7 (March 21, 2014)

Table 3.5.19

The correct headings for the second and third columns of this table are J0(t) and g(t), respectively. Previously these columns were mislabeled as g(t) and J0(t).

t J0(t) g(t)
0.0 1.00000 00000 1.00000 00000
0.5 0.93846 98072 0.93846 98072
1.0 0.76519 76866 0.76519 76865
2.0 0.22389 07791 0.22389 10326
5.0 -0.17759 67713 -0.17902 54097
10.0 -0.24593 57645 -0.07540 53543

Reported 2014-01-31 by Masataka Urago.

Table 3.5.21

The correct corner coordinates for the 9-point square, given on the last line of this table, are (±35h,±35h). Originally they were given incorrectly as (±35h,0), (±35h,0).

Diagram (xj,yj) wj R
(0,0) 1681 O(h6)
(±35h,0), (0,±35h) 1081
(±35h,±35h) 25324

Reported 2014-01-13 by Stanley Oleszczuk.

Equation (4.21.1)

4.21.1 sinu±cosu=2sin(u±14π)=±2cos(u14π)

Originally the symbol ± was missing after the second equal sign.

Reported 2012-09-27 by Dennis Heim.

Equations (4.23.34) and (4.23.35)

4.23.34 arcsinz=arcsinβ+isign(y)ln(α+(α2-1)1/2)


4.23.35 arccosz=arccosβ-isign(y)ln(α+(α2-1)1/2)

Originally the factor sign(y) was missing from the second term on the right sides of these equations. Additionally, the condition for the validity of these equations has been weakened.

Reported 2013-07-01 by Volker Thürey.

Equation (5.17.5)

5.17.5 LnG(z+1)14z2+zLnΓ(z+1)-(12z(z+1)+112)Lnz-lnA+k=1B2k+22k(2k+1)(2k+2)z2k

Originally the term zLnΓ(z+1) was incorrectly stated as zΓ(z+1).

Reported 2013-08-01 by Gergő Nemes and subsequently by Nick Jones on December 11, 2013.

Table 22.4.3

Originally a minus sign was missing in the entries for cdu and dcu in the second column (headed z+K+iK). The correct entries are -k-1nsz and -ksnz. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions sn,cn,dn, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

z+K z+K+iK z+iK z+2K z+2K+2iK z+2iK
cdu -snz -k-1nsz k-1dcz -cdz -cdz cdz
dcu -nsz -ksnz kcdz -dcz -dcz dcz

Reported 2014-02-28 by Svante Janson.

Table 22.5.2

The entry for snz at z=32(K+iK) has been corrected. The correct entry is (1+i)((1+k)1/2-i(1-k)1/2)/(2k1/2). Originally the terms (1+k)1/2 and (1-k)1/2 were given incorrectly as (1+k)1/2 and (1-k)1/2.

Similarly, the entry for dnz at z=32(K+iK) has been corrected. The correct entry is (-1+i)k1/2((1+k)1/2+i(1-k)1/2)/2. Originally the terms (1+k)1/2 and (1-k)1/2 were given incorrectly as (1+k)1/2 and (1-k)1/2

Reported 2014-02-28 by Svante Janson.

Equation (22.6.7)

22.6.7 dn(2z,k)=dn2(z,k)-k2sn2(z,k)cn2(z,k)1-k2sn4(z,k)=dn4(z,k)+k2k2sn4(z,k)1-k2sn4(z,k)

Originally the term k2sn2(z,k)cn2(z,k) was given incorrectly as k2sn2(z,k)dn2(z,k).

Reported 2014-02-28 by Svante Janson.

Table 26.8.1

Originally the Stirling number s(10,6) was given incorrectly as 6327. The correct number is 63273.

n k
0 1 2 3 4 5 6 7 8 9 10
10 0 -3 62880 10 26576 -11 72700 7 23680 -2 69325 63273 -9450 870 -45 1

Reported 2013-11-25 by Svante Janson.

Equation (31.8.5)

31.8.5 Ψ1,-1=(z2+(λ+3a+3)z+a)/z3

Originally the first term on the right side of the equation for Ψ1,-1 was z3. The correct factor is z2.

Reported 2013-07-25 by Christopher Künstler.

Equation (31.12.3)

31.12.3 d2wdz2-(γz+δ+z)dwdz+αz-qzw=0

Originally the sign in front of the second term in this equation was +. The correct sign is -.

Reported 2013-10-31 by Henryk Witek.

Equation (34.4.2)

34.4.2 {j1j2j3l1l2l3}=Δ(j1j2j3)Δ(j1l2l3)Δ(l1j2l3)Δ(l1l2j3)×s(-1)s(s+1)!(s-j1-j2-j3)!(s-j1-l2-l3)!(s-l1-j2-l3)!(s-l1-l2-j3)!×1(j1+j2+l1+l2-s)!(j2+j3+l2+l3-s)!(j3+j1+l3+l1-s)!

Originally the factor Δ(j1j2j3)Δ(j1l2l3)Δ(l1j2l3)Δ(l1l2j3) was missing in this equation.

Reported 2012-12-31 by Yu Lin.

Other Changes

Version 1.0.6 (May 6, 2013)

Several minor improvements were made affecting display and layout; primarily tracking changes to the underlying LaTeXML system.

Version 1.0.5 (October 1, 2012)

Subsection 1.2(i)

The condition for (1.2.2), (1.2.4), and (1.2.5) was corrected. These equations are true only if n is a positive integer. Previously n was allowed to be zero.

Reported 2011-08-10 by Michael Somos.

Subsection 8.17(i)

The condition for the validity of (8.17.5) is that m and n are positive integers and 0x<1. Previously, no conditions were stated.

Reported 2011-03-23 by Stephen Bourn.

Equation (10.20.14)

10.20.14 B3(0)=-959 71711 8460325 47666 37125 00000213

Originally this coefficient was given incorrectly as B3(0)=-430 99056 39368 592535 68167 34399 42500 00000213. The other coefficients in this equation have not been changed.

Reported 2012-05-11 by Antony Lee.

Equation (13.16.4)

The condition for the validity of this equation is (κ-μ)-12<0. Originally it was given incorrectly as (κ-μ)-12>0.

Subsection 14.2(ii)

Originally it was stated, incorrectly, that Qνμ(x) is real when ν,μ and x(1,). This statement is true only for Pνμ(x) and Qνμ(x).

Reported 2012-07-18 by Hans Volkmer and Howard Cohl.

Equation (21.3.4)

21.3.4 θ[α+m1β+m2](z|Ω)=e2πiαm2θ[αβ](z|Ω)

Originally the vector m2 on the right-hand side was given incorrectly as m1.

Reported 2012-08-27 by Klaas Vantournhout.

Subsection 21.10(i)

The entire original content of this subsection has been replaced by a reference.

Figures 22.3.22 and 22.3.23

The captions for these figures have been corrected to read, in part, “as a function of k2=iκ2” (instead of k2=iκ). Also, the resolution of the graph in Figure 22.3.22 was improved near κ=3.

Reported 2011-10-30 by Paul Abbott.

Equation (23.2.4)

23.2.4 (z)=1z2+w𝕃\{0}(1(z-w)2-1w2)

Originally the denominator (z-w)2 was given incorrectly as (z-w2).

Reported 2012-02-16 by James D. Walker.

Equation (24.4.26)

This equation is true only for n>0. Previously, n=0 was also allowed.

Reported 2012-05-14 by Vladimir Yurovsky.

Equation (26.12.26)

26.12.26 pp(n)(ζ(3))7/36211/36(3π)1/2n25/36exp(3(ζ(3))1/3(12n)2/3+ζ(-1)).

Originally this equation was given incorrectly as


Reported 2011-09-05 by Suresh Govindarajan.

Other Changes

Version 1.0.4 (March 23, 2012)

Several minor improvements were made affecting display of math and graphics on the web site; the software index and help files were updated.

Version 1.0.3 (Aug 29, 2011)

Equation (13.18.7)

13.18.7 W-14,±14(z2)=e12z2πzerfc(z)

Originally the left-hand side was given correctly as W-14,-14(z2); the equation is true also for W-14,+14(z2).

Other Changes

Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

Version 1.0.2 (July 1, 2011)

Several minor improvements were made affecting display on the web site; the help files were revised.

Version 1.0.1 (June 27, 2011)

Subsections 1.15(vi) and 1.15(vii)

The formulas in these subsections are valid only for x0. No conditions on x were given originally.

Reported 2010-10-18 by Andreas Kurt Richter.

Figure 10.48.5

Originally the ordinate labels 2 and 4 in this figure were placed too high.

See accompanying text

Reported 2010-11-08 by Wolfgang Ehrhardt.

Equation (14.19.2)

14.19.2 Pν-12μ(coshξ)=Γ(12-μ)π1/2(1-e-2ξ)μe(ν+(1/2))ξF(12-μ,12+ν-μ;1-2μ;1-e-2ξ),

Originally the argument to F in this equation was incorrect (e-2ξ, rather than 1-e-2ξ), and the condition on μ was too weak (μ12, rather than μ12,32,52,). Also, the factor multiplying F was rewritten to clarify the poles; originally it was Γ(1-2μ)22μΓ(1-μ)(1-e-2ξ)μe(ν+(1/2))ξ.

Reported 2010-11-02 by Alvaro Valenzuela.

Equation (17.13.3)

17.13.3 0tα-1(-tqα+β;q)(-t;q)dt=Γ(α)Γ(1-α)Γq(β)Γq(1-α)Γq(α+β),

Originally the differential was identified incorrectly as dqt; the correct differential is dt.

Reported 2011-04-08.

Table 18.9.1

The coefficient An for Cn(λ)(x) in the first row of this table originally omitted the parentheses and was given as 2n+λn+1, instead of 2(n+λ)n+1.

pn(x) An Bn Cn
Cn(λ)(x) 2(n+λ)n+1 0 n+2λ-1n+1

Reported 2010-09-16 by Kendall Atkinson.

Subsection 19.16(iii)

Originally it was implied that RC(x,y) is an elliptic integral. It was clarified that R-a(b;z) is an elliptic integral iff the stated conditions hold; originally these conditions were stated as sufficient but not necessary. In particular, RC(x,y) does not satisfy these conditions.

Reported 2010-11-23.

Table 22.5.4

Originally the limiting form for sc(z,k) in the last line of this table was incorrect (coshz, instead of sinhz).

sn(z,k) tanhz cd(z,k) 1 dc(z,k) 1 ns(z,k) cothz
cn(z,k) sechz sd(z,k) sinhz nc(z,k) coshz ds(z,k) cschz
dn(z,k) sechz nd(z,k) coshz sc(z,k) sinhz cs(z,k) cschz

Reported 2010-11-23.

Equation (22.16.14)

22.16.14 (x,k)=0sn(x,k)1-k2t21-t2dt

Originally this equation appeared with the upper limit of integration as x, rather than sn(x,k).

Reported 2010-07-08 by Charles Karney.

Equation (26.7.6)

26.7.6 B(n+1)=k=0n(nk)B(k)

Originally this equation appeared with B(n) in the summation, instead of B(k).

Reported 2010-11-07 by Layne Watson.

Equation (36.10.14)

36.10.14 3(2Ψ(E)x2-2Ψ(E)y2)+2izΨ(E)x-xΨ(E)=0

Originally this equation appeared with Ψ(H)x in the second term, rather than Ψ(E)x.

Reported 2010-04-02.

Other Changes

Version 1.0.0 (May 7, 2010)

The Handbook of Mathematical Functions was published, and the Digital Library of Mathematical Functions was released.