Digital Library of Mathematical Functions
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Errata

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 1.0.7 (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
\begin{picture}(2.4,3.0)(-1.2,-1.55)\put(0.0,0.0){\line(1,0){0.05}}\put(0.1,0.%
0){\line(1,0){0.05}}\put(0.2,0.0){\line(1,0){0.05}}\put(0.3,0.0){\line(1,0){0.%
05}}\put(0.4,0.0){\line(1,0){0.05}}\put(0.5,0.0){\line(1,0){0.05}}\put(0.6,0.0%
){\line(1,0){0.05}}\put(0.7,0.0){\line(1,0){0.05}}\put(0.8,0.0){\line(1,0){0.0%
5}}\put(0.9,0.0){\line(1,0){0.05}}
\put(0.0,0.0){\line(0,1){0.05}}\put(0.0,0.1){\line(0,1){0.05}}\put(0.0,0.2){%
\line(0,1){0.05}}\put(0.0,0.3){\line(0,1){0.05}}\put(0.0,0.4){\line(0,1){0.05}%
}\put(0.0,0.5){\line(0,1){0.05}}\put(0.0,0.6){\line(0,1){0.05}}\put(0.0,0.7){%
\line(0,1){0.05}}\put(0.0,0.8){\line(0,1){0.05}}\put(0.0,0.9){\line(0,1){0.05}%
}
\put(0.0,0.0){\line(-1,0){0.05}}\put(-0.1,0.0){\line(-1,0){0.05}}\put(-0.2,0.0%
){\line(-1,0){0.05}}\put(-0.3,0.0){\line(-1,0){0.05}}\put(-0.4,0.0){\line(-1,0%
){0.05}}\put(-0.5,0.0){\line(-1,0){0.05}}\put(-0.6,0.0){\line(-1,0){0.05}}\put%
(-0.7,0.0){\line(-1,0){0.05}}\put(-0.8,0.0){\line(-1,0){0.05}}\put(-0.9,0.0){%
\line(-1,0){0.05}}
\put(0.0,0.0){\line(0,-1){0.05}}\put(0.0,-0.1){\line(0,-1){0.05}}\put(0.0,-0.2%
){\line(0,-1){0.05}}\put(0.0,-0.3){\line(0,-1){0.05}}\put(0.0,-0.4){\line(0,-1%
){0.05}}\put(0.0,-0.5){\line(0,-1){0.05}}\put(0.0,-0.6){\line(0,-1){0.05}}\put%
(0.0,-0.7){\line(0,-1){0.05}}\put(0.0,-0.8){\line(0,-1){0.05}}\put(0.0,-0.9){%
\line(0,-1){0.05}}
\put(-1.0,1.0){\line(1,0){2.0}}
\put(-1.0,1.0){\line(0,-1){2.0}}
\put(1.0,-1.0){\line(-1,0){2.0}}
\put(1.0,-1.0){\line(0,1){2.0}}
\put(0.0,0.0){\circle*{0.15}}\put(0.7746,0.0){\circle*{0.15}}\put(-0.7746,0.0)%
{\circle*{0.15}}\put(0.0,0.7746){\circle*{0.15}}\put(0.0,-0.7746){\circle*{0.1%
5}}\put(0.7746,0.7746){\circle*{0.15}}\put(-0.7746,0.7746){\circle*{0.15}}\put%
(0.7746,-0.7746){\circle*{0.15}}\put(-0.7746,-0.7746){\circle*{0.15}}\end{picture}
(0,0) 1681 O(h6)
(±35h,0), (0,±35h) 1081
(±35h,±35h) 25324

Reported 2014-01-13 by Stanley Oleszczuk.

Equation 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


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

and

arccosz=arccosβ-sign(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


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.

u
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


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


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


2wz2-(γz+δ+z)wz+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


{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
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
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
(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
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

pp(n)(ζ(3)211n25)1/36exp(3(ζ(3)n24)1/3+ζ(-1))

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
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
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),
μ12,32,52,.

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

Originally the differential was identified incorrectly as qt; the correct differential is t.

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
(x,k)=0sn(x,k)1-k2t21-t2t

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
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
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.