Errata

♦ Version 1.0.5 (October 1, 2012) ♦ Version 1.0.4 (March 23, 2012) ♦ Version 1.0.3 (Aug 29, 2011) ♦ Version 1.0.2 (July 1, 2011) ♦ Version 1.0.1 (June 27, 2011) ♦ Version 1.0.0 (May 7, 2010) ♦

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 .

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 $0\leq x<1$ . Previously, no conditions were stated.

Reported 2011-03-23 by Stephen Bourn.

Equation 10.20.14

$B_{3}(0)=-\tfrac{959\; 71711\; 84603}{25\; 47666\; 37125\; 0 0 0 0 0}2^{{\frac{1}{3}}}$

Originally this coefficient was given incorrectly as $B_{3}(0)=-\tfrac{430\; 99056\; 39368\; 59253}{5\; 68167\; 34399\; 42500\; 0 0 0 0 0}2^{{\frac{1}{3}}}$ . 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 $\realpart{(\kappa-\mu)-\tfrac{1}{2}}<0$ . Originally it was given incorrectly as $\realpart{(\kappa-\mu)-\tfrac{1}{2}}>0$ .

Subsection 14.2(ii)

Originally it was stated, incorrectly, that $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ is real when $\nu,\mu\in\Real$ and $x\in(1,\infty)$ . This statement is true only for $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ .

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

Equation 21.3.4

$\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{\boldsymbol{{\beta}}+\mathbf{m}_{2}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$

Originally the vector $\mathbf{m}_{2}$ was given incorrectly as $\mathbf{m}_{1}$ .

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 $k^{2}=i\kappa^{2}$ ” (instead of $k^{2}=i\kappa$ ). Also, the resolution of the graph in Figure 22.3.22 was improved near $\kappa=3$ .

Reported 2011-10-30 by Paul Abbott.

Equation 23.2.4

$\mathop{\wp\/}\nolimits\!\left(z\right)=\frac{1}{z^{2}}+\sum _{{w\in\mathbb{L}\setminus\{ 0\}}}\left(\frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right)$

Originally the denominator $(z-w)^{2}$ was given incorrectly as $(z-w^{2})$ .

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

$\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)\sim\frac{\left(\mathop{\zeta\/}\nolimits\!\left(3\right)\right)^{{7/36}}}{2^{{11/36}}(3\pi)^{{1/2}}n^{{25/36}}}\*\mathop{\exp\/}\nolimits\left(3\left(\mathop{\zeta\/}\nolimits\!\left(3\right)\right)^{{1/3}}\left(\tfrac{1}{2}n\right)^{{2/3}}+{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-1\right)\right)$

Originally this equation was given incorrectly as

$\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)\sim\left(\frac{\mathop{\zeta\/}\nolimits\!\left(3\right)}{2^{{11}}n^{{25}}}\right)^{{1/36}}\*\mathop{\exp\/}\nolimits\left(3\left(\frac{\mathop{\zeta\/}\nolimits\!\left(3\right)n^{2}}{4}\right)^{{1/3}}+{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-1\right)\right)$

Reported 2011-09-05 by Suresh Govindarajan.

Other Changes

Bibliographic citations were added in §§5.5(iii), 5.6(i), 5.10, 5.21, 7.13(ii), 10.19(iii), 10.21(i), 10.21(iv), 10.21(xiii), 10.21(xiv), 10.42, 10.46, 10.74(vii), 13.8(ii), 13.9(i), 13.9(ii), 13.11, 13.29(iv), 14.11, 15.13, 15.19(i), 17.18, 18.16(ii), 18.16(iv), 18.26(v), 19.12, 19.36(iv), 20.7(i), 20.7(ii), 25.11(iv), 25.18(i), 28.24, 28.34(ii), 29.20(i), 31.17(ii), and 32.17.
Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).
The upper and lower bounds given in Equations (18.16.12) and (18.16.13) have been replaced with stronger bounds.
Textual clarifications were made in §§1.5(ii), 7.13(ii), 15.6, 19.12, 30.13(i), 30.14(i), 20.7(iv), and 31.17(ii).
Other minor changes were made in the bibliography and index.

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: $\mathop{W_{{-\frac{1}{4},\pm\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)=e^{{\frac{1}{2}z^{2}}}\sqrt{\pi z}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z\right)$

Originally the left-hand side was given correctly as $\mathop{W_{{-\frac{1}{4},-\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)$ ; the equation is true also for $\mathop{W_{{-\frac{1}{4},+\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)$ .
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 $x\geq 0$ . 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.

Reported 2010-11-08 by Wolfgang Ehrhardt.

Equation 14.19.2

$\mathop{P^{{\mu}}_{{\nu-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu\right)}{\pi^{{1/2}}\left(1-e^{{-2\xi}}\right)^{\mu}e^{{(\nu+(1/2))\xi}}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{{-2\xi}}\right),$ $\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$ .

Originally the argument to $\mathop{\mathbf{F}\/}\nolimits$ in this equation was incorrect ( $e^{{-2\xi}}$ , rather than $1-e^{{-2\xi}}$ ), and the condition on $\mu$ was too weak ( $\mu\neq\frac{1}{2}$ , rather than $\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$ ). Also, the factor multiplying $\mathop{\mathbf{F}\/}\nolimits$ was rewritten to clarify the poles; originally it was $\frac{\mathop{\Gamma\/}\nolimits\!\left(1-2\mu\right)2^{{2\mu}}}{\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)\left(1-e^{{-2\xi}}\right)^{\mu}e^{{(\nu+(1/2))\xi}}}$ .

Reported 2010-11-02 by Alvaro Valenzuela.

Equation 17.13.3

$\int _{0}^{\infty}t^{{\alpha-1}}\frac{\left(-tq^{{\alpha+\beta}};q\right)_{{\infty}}}{\left(-t;q\right)_{{\infty}}}dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(1-\alpha\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\beta\right)}{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(1-\alpha\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\alpha+\beta\right)},$

Originally the differential was identified incorrectly as ${d}_{q}t$ ; the correct differential is $dt$ .

Reported 2011-04-08.

Table 18.9.1

The coefficient $A_{n}$ for $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ in the first row of this table originally omitted the parentheses and was given as $\frac{2n+\lambda}{n+1}$ , instead of $\frac{2(n+\lambda)}{n+1}$ .

$p_{n}(x)$	$A_{n}$	$B_{n}$	$C_{n}$
$\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$	$\frac{2(n+\lambda)}{n+1}$	0	$\frac{n+2\lambda-1}{n+1}$
$\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$	$2-\delta _{{n,0}}$	0	1
$\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$	2	0	1
$\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$	$4-2\delta _{{n,0}}$	$-2+\delta _{{n,0}}$	1
$\mathop{U^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$	4	−2	1
$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$	$\frac{2n+1}{n+1}$	0	$\frac{n}{n+1}$
$\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$	$\frac{4n+2}{n+1}$	$-\frac{2n+1}{n+1}$	$\frac{n}{n+1}$
$\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$	$-\frac{1}{n+1}$	$\frac{2n+\alpha+1}{n+1}$	$\frac{n+\alpha}{n+1}$
$\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$	2	0	$2n$
$\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$	1	0	$n$

Reported 2010-09-16 by Kendall Atkinson.

Subsection 19.16(iii)

Originally it was implied that $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ is an elliptic integral. It was clarified that $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ is an elliptic integral iff the stated conditions hold; originally these conditions were stated as sufficient but not necessary. In particular, $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ does not satisfy these conditions.

Reported 2010-11-23.

Table 22.5.4

Originally the limiting form for $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ in the last line of this table was incorrect ( $\mathop{\cosh\/}\nolimits z$ , instead of $\mathop{\sinh\/}\nolimits z$ ).

$\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\tanh\/}\nolimits z$	$\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ $\;\to\;$	1	$\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ $\;\to\;$	1	$\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\coth\/}\nolimits z$
$\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\mathrm{sech}\/}\nolimits z$	$\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\sinh\/}\nolimits z$	$\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\cosh\/}\nolimits z$	$\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\mathrm{csch}\/}\nolimits z$
$\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\mathrm{sech}\/}\nolimits z$	$\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\cosh\/}\nolimits z$	$\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\sinh\/}\nolimits z$	$\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ $\;\to\;$	$\mathop{\mathrm{csch}\/}\nolimits z$

Reported 2010-11-23.

Equation 22.16.14

$\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)=\int _{0}^{{\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)}}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt$

Originally this equation appeared with the upper limit of integration as $x$ , rather than $\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)$ .

Reported 2010-07-08 by Charles Karney.

Equation 26.7.6

$\mathop{B\/}\nolimits\!\left(n+1\right)=\sum _{{k=0}}^{n}\binom{n}{k}\mathop{B\/}\nolimits\!\left(k\right)$

Originally this equation appeared with $\mathop{B\/}\nolimits\!\left(n\right)$ in the summation, instead of $\mathop{B\/}\nolimits\!\left(k\right)$ .

Reported 2010-11-07 by Layne Watson.

Equation 36.10.14

$3\left(\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{{\partial x}^{2}}-\frac{{\partial}^{2}\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{{\partial y}^{2}}\right)+2iz\frac{\partial\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{\partial x}-x\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits=0$

Originally this equation appeared with $\frac{\partial\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits}{\partial x}$ in the second term, rather than $\frac{\partial\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits}{\partial x}$ .

Reported 2010-04-02.

Other Changes

The definition of the notation $F(z_{0}e^{{2k\pi i}})$ was added in Common Notations and Definitions.
Clarifications were made in §§5.18, 7.1, 9.2(iii), 10.15, 10.38, 14.13, 15.8(i), 15.10(i), 16.11(ii), 19.2(iv), 19.16(ii), 19.16(iii), 22.16(ii), 27.16.
Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.
The general references for each chapter were inserted under the i-symbol on the chapter title pages. Originally these appeared only in the References sections of the individual chapters in the Handbook.
The definition of $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ was revised in Notations *.
Additions and revisions were made in the Cross Index for Computing Special Functions.

Version 1.0.0 (May 7, 2010)

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