# 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 .

# Version 1.0.8 (April 25, 2014)

Equation (22.19.2)
 $\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\theta(t)\right)=\mathop{\sin\/}% \nolimits\!\left(\frac{1}{2}\alpha\right)\mathop{\mathrm{sn}\/}\nolimits\left(% t+K,\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\alpha\right)\right)$

Originally the $K$ was missing in the $t+K$.

Equation (22.19.3)
 $\theta(t)=2\mathop{\mathrm{am}\/}\nolimits\left(t\sqrt{E/2},\sqrt{2/E}\right)$

Originally the factor $\sqrt{E/2}$ was missing in the $t\sqrt{E/2}$.

Other Changes

• 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 $\mathop{J_{0}\/}\nolimits\!\left(t\right)$ and $g(t)$, respectively. Previously these columns were mislabeled as $g(t)$ and $\mathop{J_{0}\/}\nolimits\!\left(t\right)$.

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 $(\pm\sqrt{\tfrac{3}{5}}h,\pm\sqrt{\tfrac{3}{5}}h)$. Originally they were given incorrectly as $(\pm\sqrt{\tfrac{3}{5}}h,0)$, $(\pm\sqrt{\tfrac{3}{5}}h,0)$.

Reported 2014-01-13 by Stanley Oleszczuk.

Equation (4.21.1)

 $\mathop{\sin\/}\nolimits u\pm\mathop{\cos\/}\nolimits u=\sqrt{2}\mathop{\sin\/% }\nolimits\!\left(u\pm\tfrac{1}{4}\pi\right)=\pm\sqrt{2}\mathop{\cos\/}% \nolimits\!\left(u\mp\tfrac{1}{4}\pi\right)$

Originally the symbol $\pm$ was missing after the second equal sign.

Reported 2012-09-27 by Dennis Heim.

Equations (4.23.34) and (4.23.35)

 $\mathop{\mathrm{arcsin}\/}\nolimits z=\mathop{\mathrm{arcsin}\/}\nolimits\beta% +i\mathop{\mathrm{sign}\/}\nolimits\!\left(y\right)\mathop{\ln\/}\nolimits\!% \left(\alpha+(\alpha^{2}-1)^{1/2}\right)$

and

 $\mathop{\mathrm{arccos}\/}\nolimits z=\mathop{\mathrm{arccos}\/}\nolimits\beta% -i\mathop{\mathrm{sign}\/}\nolimits\!\left(y\right)\mathop{\ln\/}\nolimits\!% \left(\alpha+(\alpha^{2}-1)^{1/2}\right)$

Originally the factor $\mathop{\mathrm{sign}\/}\nolimits\!\left(y\right)$ 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)

 $\mathop{\mathrm{Ln}\/}\nolimits\mathop{G\/}\nolimits\!\left(z+1\right)\sim% \tfrac{1}{4}z^{2}+z\mathop{\mathrm{Ln}\/}\nolimits\mathop{\Gamma\/}\nolimits\!% \left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\mathop{\mathrm{% Ln}\/}\nolimits z-\mathop{\ln\/}\nolimits A+\sum_{k=1}^{\infty}\frac{\mathop{B% _{2k+2}\/}\nolimits}{2k(2k+1)(2k+2)z^{2k}}$

Originally the term $z\mathop{\mathrm{Ln}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(z+1\right)$ was incorrectly stated as $z\mathop{\Gamma\/}\nolimits\!\left(z+1\right)$.

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 $\mathop{\mathrm{cd}\/}\nolimits u$ and $\mathop{\mathrm{dc}\/}\nolimits u$ in the second column (headed $z+K+iK^{\prime}$). The correct entries are $-k^{-1}\mathop{\mathrm{ns}\/}\nolimits z$ and $-k\mathop{\mathrm{sn}\/}\nolimits z$. 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 $\mathop{\mathrm{sn}\/}\nolimits,\mathop{\mathrm{cn}\/}\nolimits,\mathop{% \mathrm{dn}\/}\nolimits$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

Reported 2014-02-28 by Svante Janson.

Table 22.5.2

The entry for $\mathop{\mathrm{sn}\/}\nolimits z$ at $z=\frac{3}{2}(K+iK^{\prime})$ has been corrected. The correct entry is $(1+i)((1+k^{\prime})^{1/2}-i(1-k^{\prime})^{1/2})/(2k^{1/2})$. Originally the terms $(1+k^{\prime})^{1/2}$ and $(1-k^{\prime})^{1/2}$ were given incorrectly as $(1+k)^{1/2}$ and $(1-k)^{1/2}$.

Similarly, the entry for $\mathop{\mathrm{dn}\/}\nolimits z$ at $z=\frac{3}{2}(K+iK^{\prime})$ has been corrected. The correct entry is $(-1+i){k^{\prime}}^{1/2}((1+k)^{1/2}+i(1-k)^{1/2})/2$. Originally the terms $(1+k)^{1/2}$ and $(1-k)^{1/2}$ were given incorrectly as $(1+k^{\prime})^{1/2}$ and $(1-k^{\prime})^{1/2}$

Reported 2014-02-28 by Svante Janson.

Equation (22.6.7)

 $\mathop{\mathrm{dn}\/}\nolimits\left(2z,k\right)=\frac{{\mathop{\mathrm{dn}\/}% \nolimits^{2}}\left(z,k\right)-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{2}}\left% (z,k\right){\mathop{\mathrm{cn}\/}\nolimits^{2}}\left(z,k\right)}{1-k^{2}{% \mathop{\mathrm{sn}\/}\nolimits^{4}}\left(z,k\right)}=\frac{{\mathop{\mathrm{% dn}\/}\nolimits^{4}}\left(z,k\right)+k^{2}{k^{\prime}}^{2}{\mathop{\mathrm{sn}% \/}\nolimits^{4}}\left(z,k\right)}{1-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{4}% }\left(z,k\right)}$

Originally the term $k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{2}}\left(z,k\right){\mathop{\mathrm{cn}% \/}\nolimits^{2}}\left(z,k\right)$ was given incorrectly as $k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{2}}\left(z,k\right){\mathop{\mathrm{dn}% \/}\nolimits^{2}}\left(z,k\right)$.

Reported 2014-02-28 by Svante Janson.

Table 26.8.1

Originally the Stirling number $\mathop{s\/}\nolimits\!\left(10,6\right)$ was given incorrectly as 6327. The correct number is 63273.

Reported 2013-11-25 by Svante Janson.

Equation (31.8.5)

 $\Psi_{1,-1}=\left(z^{2}+(\lambda+3a+3)z+a\right)/z^{3}$

Originally the first term on the right side of the equation for $\Psi_{1,-1}$ was $z^{3}$. The correct factor is $z^{2}$.

Reported 2013-07-25 by Christopher Künstler.

Equation (31.12.3)

 $\frac{{d}^{2}w}{{dz}^{2}}-\left(\frac{\gamma}{z}+\delta+z\right)\frac{dw}{dz}+% \frac{\alpha z-q}{z}w=0$

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)

 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!}$

Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation.

Reported 2012-12-31 by Yu Lin.

Other Changes

• Equations (4.45.8) and (4.45.9) have been replaced with equations that are better for numerically computing $\mathop{\mathrm{arctan}\/}\nolimits x$.

• A new subsection 13.29(v) has been added to cover computation of confluent hypergeometric functions by continued fractions.

• A new subsection 14.5(vi) containing the values of Legendre and Ferrers functions for degree $\nu=2$ has been added.

• Subsection 14.18(iii) has been altered to identify Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

• A new subsection 15.19(v) has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

• Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

• Cross-references have been added in §§1.2(i), 10.19(iii), 10.23(ii), 17.2(iii), 18.15(iii), 19.2(iv), 19.16(i).

• Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).

• Entries for the Sage computational system have been updated in the Software Index.

• The default document format for DLMF is now HTML5 which includes MathML providing better accessibility and display of mathematics.

• All interactive 3D graphics on the DLMF website have been recast using WebGL and X3DOM, improving portability and performance; WebGL it is now the default format.

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

Reported 2011-03-23 by Stephen Bourn.

Equation (10.20.14)
 $B_{3}(0)=-\tfrac{959\;71711\;84603}{25\;47666\;37125\;00000}2^{\frac{1}{3}}$

Originally this coefficient was given incorrectly as $B_{3}(0)=-\tfrac{430\;99056\;39368\;59253}{5\;68167\;34399\;42500\;00000}2^{% \frac{1}{3}}$. 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 $\mathbf{m}_{2}$ on the right-hand side 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.

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

# 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 $\frac{2(n+\lambda)}{n+1}$.

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 $\mathop{\sinh\/}\nolimits z$).

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

# Version 1.0.0 (May 7, 2010)

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