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.
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.Reported 20151112 by James W. Pitman.
 Equation (9.7.17)

Originally the constraint, $$, was written incorrectly as $\frac{2}{3}\pi \le \mathrm{ph}z\le \pi $. Also, the equation was reformatted to display the constraints in the equation instead of in the text.Reported 20141105 by Gergő Nemes.
 Equation (10.32.13)

Originally the constraint, $$, was incorrectly written as, $$.Reported 20150520 by Richard Paris.
 Equation (10.40.12)

Originally the third constraint $$ was incorrectly written as $\pi \le \mathrm{ph}z\le \frac{3}{2}\pi $.Reported 20141105 by Gergő Nemes.
 Equation (23.18.7)

23.18.7 $$s(d,c)=\sum _{r=1}^{c1}\frac{r}{c}\left(\frac{dr}{c}\lfloor \frac{dr}{c}\rfloor \frac{1}{2}\right),$$ $c>0$. Originally the sum ${\sum}_{r=1}^{c1}$ was written with an additional condition on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.
Reported 20151005 by Howard Cohl and Tanay Wakhare.
 Equations (28.28.21) and (28.28.22)

28.28.21 $$\begin{array}{l}\frac{4}{\pi}{\int}_{0}^{\pi /2}{\mathcal{C}}_{2\mathrm{\ell}+1}^{(j)}(2hR)\mathrm{cos}\left((2\mathrm{\ell}+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,{h}^{2})dt\\ \phantom{\rule{2em}{0ex}}={(1)}^{\mathrm{\ell}+m}{A}_{2\mathrm{\ell}+1}^{2m+1}({h}^{2}){\mathrm{Mc}}_{2m+1}^{(j)}(z,h),\end{array}$$ 28.28.22 $$\begin{array}{l}\frac{4}{\pi}{\int}_{0}^{\pi /2}{\mathcal{C}}_{2\mathrm{\ell}+1}^{(j)}(2hR)\mathrm{sin}\left((2\mathrm{\ell}+1)\varphi \right){\mathrm{se}}_{2m+1}(t,{h}^{2})dt\\ \phantom{\rule{2em}{0ex}}={(1)}^{\mathrm{\ell}+m}{B}_{2\mathrm{\ell}+1}^{2m+1}({h}^{2}){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),\end{array}$$ Originally the prefactor $\frac{4}{\pi}$ and upper limit of integration $\pi /2$ in these two equations were given incorrectly as $\frac{2}{\pi}$ and $\pi $.
Reported 20150520 by Ruslan Kabasayev
 Other Changes

 –

–
It was reported by Nico Temme on 20150228 that the asymptotic formula for $\mathrm{Ln}\mathrm{\Gamma}\left(z+h\right)$ given in (5.11.8) is valid for $h\phantom{\rule{veryverythickmathspace}{0ex}}(\in \u2102)$; originally it was unnecessarily restricted to $[0,1]$.

–
A new paragraph with several new equations and a new reference has been added at the end of §13.8(iii) to provide asymptotic expansions for Kummer functions $U(a,b,z)$ and $\mathbf{M}(a,b,z)$ as $a\to \mathrm{\infty}$ in $\mathrm{ph}a\le \pi \delta $ and $b$ and $z$ fixed.

–
Because of the use of the $O$ order symbol on the righthand side, the asymptotic expansion (18.15.22) for the generalized Laguerre polynomial ${L}_{n}^{(\alpha )}\left(\nu x\right)$ was rewritten as an equality.

–
The entire section §27.20 was replaced.

–
Bibliographic citations have been added or modified in §§2.4(v), 2.4(vi), 2.9(iii), 5.11(i), 5.11(ii), 5.17, 9.9(i), 10.22(v), 10.37, 11.6(iii), 11.9(iii), 12.9(i), 13.8(ii), 13.11, 14.15(i), 14.15(iii), 15.12(iii), 15.14, 16.11(ii), 16.13, 18.15(vi), 20.7(viii), 24.11, 24.16(i), 26.8(vii), 33.12(i), and 33.12(ii).
 –
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 $p\phantom{\rule{veryverythickmathspace}{0ex}}(\ne 0)$ and $q$ be real constants and
4.43.1 $A$ $={\left(\frac{4}{3}p\right)}^{1/2},$ $B$ $={\left(\frac{4}{3}p\right)}^{1/2}.$ The roots of
4.43.2 $${z}^{3}+pz+q=0$$ are:

(a)
$A\mathrm{sin}a$, $A\mathrm{sin}\left(a+\frac{2}{3}\pi \right)$, and $A\mathrm{sin}\left(a+\frac{4}{3}\pi \right)$, with $\mathrm{sin}\left(3a\right)=4q/{A}^{3}$, when $4{p}^{3}+27{q}^{2}\le 0$.

(b)
$A\mathrm{cosh}a$, $A\mathrm{cosh}\left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\mathrm{cosh}\left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\mathrm{cosh}\left(3a\right)=4q/{A}^{3}$, when $$, $$, and $4{p}^{3}+27{q}^{2}>0$.

(c)
$B\mathrm{sinh}a$, $B\mathrm{sinh}\left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\mathrm{sinh}\left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\mathrm{sinh}\left(3a\right)=4q/{B}^{3}$, 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 20141031 by Masataka Urago.

(a)
 Equation (9.10.18)

9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3{z}^{5/4}{\mathrm{e}}^{(2/3){z}^{3/2}}}{4\pi}{\int}_{0}^{\mathrm{\infty}}\frac{{t}^{3/4}{\mathrm{e}}^{(2/3){t}^{3/2}}\mathrm{Ai}\left(t\right)}{{z}^{3/2}+{t}^{3/2}}dt.$$ The original equation taken from Schulten et al. (1979) was incorrect.
Reported 20150320 by Walter Gautschi.
 Equation (9.10.19)

9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3{x}^{5/4}{\mathrm{e}}^{(2/3){x}^{3/2}}}{2\pi}{\u2a0d}_{0}^{\mathrm{\infty}}\frac{{t}^{3/4}{\mathrm{e}}^{(2/3){t}^{3/2}}\mathrm{Ai}\left(t\right)}{{x}^{3/2}{t}^{3/2}}dt.$$ The original equation taken from Schulten et al. (1979) was incorrect.
Reported 20150320 by Walter Gautschi.
 Equation (10.17.14)

10.17.14 $$\left{R}_{\mathrm{\ell}}^{\pm}(\nu ,z)\right\le 2{a}_{\mathrm{\ell}}(\nu ){\mathcal{V}}_{z,\pm \mathrm{i}\mathrm{\infty}}\left({t}^{\mathrm{\ell}}\right)\mathrm{exp}\left({\nu}^{2}\frac{1}{4}{\mathcal{V}}_{z,\pm \mathrm{i}\mathrm{\infty}}\left({t}^{1}\right)\right)$$ Originally the factor ${\mathcal{V}}_{z,\pm \mathrm{i}\mathrm{\infty}}\left({t}^{1}\right)$ in the argument to the exponential was written incorrectly as ${\mathcal{V}}_{z,\pm \mathrm{i}\mathrm{\infty}}\left({t}^{\mathrm{\ell}}\right)$.
Reported 20140927 by Gergő Nemes.
 Equation (10.19.11)

10.19.11 $${Q}_{3}\left(a\right)=\frac{549}{28000}{a}^{8}\frac{\mathrm{1\hspace{0.33em}10767}}{\mathrm{6\hspace{0.33em}93000}}{a}^{5}+\frac{79}{12375}{a}^{2}$$ Originally the first term on the righthand side of this equation was written incorrectly as $\frac{549}{28000}{a}^{8}$.
Reported 20150316 by Svante Janson.
 Equation (13.2.7)

13.2.7 $$U(m,b,z)={(1)}^{m}{\left(b\right)}_{m}M(m,b,z)={(1)}^{m}\sum _{s=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{s}\right){\left(b+s\right)}_{ms}{(z)}^{s}$$ The equality $U(m,b,z)={(1)}^{m}{\left(b\right)}_{m}M(m,b,z)$ 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 20150210 by Adri Olde Daalhuis.
 Equation (13.2.8)

13.2.8 $$U(a,a+n+1,z)=\frac{{(1)}^{n}{\left(1an\right)}_{n}}{{z}^{a+n}}M(n,1an,z)={z}^{a}\sum _{s=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{s}\right){\left(a\right)}_{s}{z}^{s}$$ The equality $U(a,a+n+1,z)=\frac{{(1)}^{n}{\left(1an\right)}_{n}}{{z}^{a+n}}M(n,1an,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.
Reported 20150210 by Adri Olde Daalhuis.
 Equation (13.2.10)

13.2.10 $$U(m,n+1,z)={(1)}^{m}{\left(n+1\right)}_{m}M(m,n+1,z)={(1)}^{m}\sum _{s=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{s}\right){\left(n+s+1\right)}_{ms}{(z)}^{s}$$ The equality $U(m,n+1,z)={(1)}^{m}{\left(n+1\right)}_{m}M(m,n+1,z)$ 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 $a=m,m=0,1,2,\mathrm{\dots}$ has been introduced.
Reported 20150210 by Adri Olde Daalhuis.
 Equation (18.33.3)

18.33.3 $${\varphi}_{n}^{*}(z)={z}^{n}\overline{{\varphi}_{n}({\overline{z}}^{1})}={\kappa}_{n}+\sum _{\mathrm{\ell}=1}^{n}{\overline{\kappa}}_{n,n\mathrm{\ell}}{z}^{\mathrm{\ell}}$$ Originally this equation was written incorrectly as ${\varphi}_{n}^{*}(z)={\kappa}_{n}{z}^{n}+{\sum}_{\mathrm{\ell}=1}^{n}{\overline{\kappa}}_{n,n\mathrm{\ell}}{z}^{n\mathrm{\ell}}$. Also, the equality ${\varphi}_{n}^{*}(z)={z}^{n}\overline{{\varphi}_{n}({\overline{z}}^{1})}$ has been added.
Reported 20141003 by Roderick Wong.
 Equation (34.7.4)

34.7.4 $$\begin{array}{l}\left(\begin{array}{ccc}\hfill {j}_{13}\hfill & \hfill {j}_{23}\hfill & \hfill {j}_{33}\hfill \\ \hfill {m}_{13}\hfill & \hfill {m}_{23}\hfill & \hfill {m}_{33}\hfill \end{array}\right)\left\{\begin{array}{ccc}\hfill {j}_{11}\hfill & \hfill {j}_{12}\hfill & \hfill {j}_{13}\hfill \\ \hfill {j}_{21}\hfill & \hfill {j}_{22}\hfill & \hfill {j}_{23}\hfill \\ \hfill {j}_{31}\hfill & \hfill {j}_{32}\hfill & \hfill {j}_{33}\hfill \end{array}\right\}\\ \phantom{\rule{2em}{0ex}}=\sum _{{m}_{r1},{m}_{r2},r=1,2,3}\begin{array}{l}\left(\begin{array}{ccc}\hfill {j}_{11}\hfill & \hfill {j}_{12}\hfill & \hfill {j}_{13}\hfill \\ \hfill {m}_{11}\hfill & \hfill {m}_{12}\hfill & \hfill {m}_{13}\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill {j}_{21}\hfill & \hfill {j}_{22}\hfill & \hfill {j}_{23}\hfill \\ \hfill {m}_{21}\hfill & \hfill {m}_{22}\hfill & \hfill {m}_{23}\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill {j}_{31}\hfill & \hfill {j}_{32}\hfill & \hfill {j}_{33}\hfill \\ \hfill {m}_{31}\hfill & \hfill {m}_{32}\hfill & \hfill {m}_{33}\hfill \end{array}\right)\\ \phantom{\rule{2em}{0ex}}\times \left(\begin{array}{ccc}\hfill {j}_{11}\hfill & \hfill {j}_{21}\hfill & \hfill {j}_{31}\hfill \\ \hfill {m}_{11}\hfill & \hfill {m}_{21}\hfill & \hfill {m}_{31}\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill {j}_{12}\hfill & \hfill {j}_{22}\hfill & \hfill {j}_{32}\hfill \\ \hfill {m}_{12}\hfill & \hfill {m}_{22}\hfill & \hfill {m}_{32}\hfill \end{array}\right)\end{array}\end{array}$$ Originally the third $\mathit{3}j$ symbol in the summation was written incorrectly as $\left(\begin{array}{ccc}\hfill {j}_{31}\hfill & \hfill {j}_{32}\hfill & \hfill {j}_{33}\hfill \\ \hfill {m}_{13}\hfill & \hfill {m}_{23}\hfill & \hfill {m}_{33}\hfill \end{array}\right).$
Reported 20150119 by YanRui Liu.
 Other Changes


–
To increase the regions of validity (5.9.10), (5.9.11), (5.10.1), (5.11.1), and (5.11.8), the logarithms of the gamma function that appears on their lefthand sides have all been changed to $\mathrm{Ln}\mathrm{\Gamma}(\cdot )$, where $\mathrm{Ln}$ is the general logarithm. Originally $\mathrm{ln}\mathrm{\Gamma}(\cdot )$ was used, where $\mathrm{ln}$ is the principal branch of the logarithm. These changes were recommended by Philippe Spindel on 20150206.
 –

–
A note was added after (22.20.5) to deal with cases when computation of $\mathrm{dn}(x,k)$ becomes numerically unstable near $x=K$.

–
The spelling of the name Delannoy was corrected in several places in §26.6. Previously it was mispelled as Dellanoy.

–
For consistency of notation across all chapters, the notation for logarithm has been changed to $\mathrm{ln}$ from $\mathrm{log}$ throughout Chapter 27.

–
Bibliographic citations have been added or modified in §§2.4(vi), 3.8(v), 5.6(i), 5.10, 5.11(i), 5.11(ii), 5.18(ii), 7.21, 8.10, 10.21(ix), 10.45, 10.74(vi), 11.7(v), 13.7(iii), 14.17(iii), 14.20(ix), 14.28(ii), 14.32, 15.8(v), 15.13, 15.19(i), 16.6, 16.13, 17.6(ii), 17.7(iii), 18.1(iii), 18.3, 18.15(iv) and 18.24.

–
Version 1.0.9 (August 29, 2014)
 Equation (9.6.26)

9.6.26 $${\mathrm{Bi}}^{\prime}\left(z\right)=\frac{{3}^{1/6}}{\mathrm{\Gamma}\left(\frac{1}{3}\right)}{\mathrm{e}}^{\zeta}{}_{1}F_{1}(\frac{1}{6};\frac{1}{3};2\zeta )+\frac{{3}^{7/6}}{{2}^{7/3}\mathrm{\Gamma}\left(\frac{2}{3}\right)}{\zeta}^{4/3}{\mathrm{e}}^{\zeta}{}_{1}F_{1}(\frac{7}{6};\frac{7}{3};2\zeta )$$ Originally the second occurrence of the function ${}_{1}F_{1}$ was given incorrectly as ${}_{1}F_{1}(\frac{7}{6};\frac{7}{3};\zeta )$.
Reported 20140521 by Hanyou Chu.
 Equation (22.19.6)

22.19.6 $$x(t)=\mathrm{cn}(t\sqrt{1+2\eta},k)$$ Originally the term $\sqrt{1+2\eta}$ was given incorrectly as $\sqrt{1+\eta}$ in this equation and in the line above. Additionally, for improved clarity, the modulus $k=1/\sqrt{2+{\eta}^{1}}$ has been defined in the line above.
Reported 20140502 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 $$. Previously it was $\sqrt{1/\beta}\le a\le \sqrt{2/\beta}$. Second, the correct period of the oscillations is $2K\left(k\right)/\sqrt{\eta}$. Previously it was given incorrectly as $4K\left(k\right)/\sqrt{\eta}$.
Reported 20140502 by Svante Janson.
 Equation (34.3.7)

34.3.7 $$\left(\begin{array}{ccc}\hfill {j}_{1}\hfill & \hfill {j}_{2}\hfill & \hfill {j}_{3}\hfill \\ \hfill {j}_{1}\hfill & \hfill {j}_{1}{m}_{3}\hfill & \hfill {m}_{3}\hfill \end{array}\right)={(1)}^{{j}_{1}{j}_{2}{m}_{3}}{\left(\frac{(2{j}_{1})!({j}_{1}+{j}_{2}+{j}_{3})!({j}_{1}+{j}_{2}+{m}_{3})!({j}_{3}{m}_{3})!}{({j}_{1}+{j}_{2}+{j}_{3}+1)!({j}_{1}{j}_{2}+{j}_{3})!({j}_{1}+{j}_{2}{j}_{3})!({j}_{1}+{j}_{2}{m}_{3})!({j}_{3}+{m}_{3})!}\right)}^{\frac{1}{2}}$$ In the original equation the prefactor of the above 3j symbol read ${(1)}^{{j}_{2}+{j}_{3}+{m}_{3}}$. It is now replaced by its correct value ${(1)}^{{j}_{1}{j}_{2}{m}_{3}}$.
Reported 20140612 by James Zibin.
 Other Changes

 –

–
The Wronskian (14.2.7) was generalized to include both associated Legendre and Ferrers functions.

–
A crossreference has been added in §15.9(iv).
 –
 –

–
An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).
Version 1.0.8 (April 25, 2014)
 Equation (22.19.2)

22.19.2 $$\mathrm{sin}\left(\frac{1}{2}\theta (t)\right)=\mathrm{sin}\left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\mathrm{sin}\left(\frac{1}{2}\alpha \right))$$ Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.
Reported 20140305 by Svante Janson.
 Equation (22.19.3)

22.19.3 $$\theta (t)=2\mathrm{am}(t\sqrt{E/2},\sqrt{2/E})$$ Originally the first argument to the function $\mathrm{am}$ was given incorrectly as $t$. The correct argument is $t\sqrt{E/2}$.
Reported 20140305 by Svante Janson.
 Other Changes

Version 1.0.7 (March 21, 2014)
 Table 3.5.19

The correct headings for the second and third columns of this table are ${J}_{0}\left(t\right)$ and $g(t)$, respectively. Previously these columns were mislabeled as $g(t)$ and ${J}_{0}\left(t\right)$.
$t$ ${J}_{0}(t)$ $g(t)$ 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 20140131 by Masataka Urago.
 Table 3.5.21

The correct corner coordinates for the 9point square, given on the last line of this table, are $(\pm \sqrt{\frac{3}{5}}h,\pm \sqrt{\frac{3}{5}}h)$. Originally they were given incorrectly as $(\pm \sqrt{\frac{3}{5}}h,0)$, $(\pm \sqrt{\frac{3}{5}}h,0)$.
Diagram $({x}_{j},{y}_{j})$ ${w}_{j}$ $R$ ⋮ $(0,0)$ $\frac{16}{81}$ $O\left({h}^{6}\right)$ $(\pm \sqrt{\frac{3}{5}}h,0)$, $(0,\pm \sqrt{\frac{3}{5}}h)$ $\frac{10}{81}$ $(\pm \sqrt{\frac{3}{5}}h,\pm \sqrt{\frac{3}{5}}h)$ $\frac{25}{324}$ Reported 20140113 by Stanley Oleszczuk.
 Equation (4.21.1)

4.21.1 $$\mathrm{sin}u\pm \mathrm{cos}u=\sqrt{2}\mathrm{sin}\left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\mathrm{cos}\left(u\mp \frac{1}{4}\pi \right)$$ Originally the symbol $\pm $ was missing after the second equal sign.
Reported 20120927 by Dennis Heim.
 Equations (4.23.34) and (4.23.35)

4.23.34 $$\mathrm{arcsin}z=\mathrm{arcsin}\beta +\mathrm{i}\mathrm{sign}\left(y\right)\mathrm{ln}\left(\alpha +{({\alpha}^{2}1)}^{1/2}\right)$$ and
4.23.35 $$\mathrm{arccos}z=\mathrm{arccos}\beta \mathrm{i}\mathrm{sign}\left(y\right)\mathrm{ln}\left(\alpha +{({\alpha}^{2}1)}^{1/2}\right)$$ Originally the factor $\mathrm{sign}\left(y\right)$ 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 20130701 by Volker Thürey.
 Equation (5.17.5)

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \begin{array}{l}\frac{1}{4}{z}^{2}+z\mathrm{Ln}\mathrm{\Gamma}\left(z+1\right)\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\mathrm{Ln}z\mathrm{ln}A\\ \phantom{\rule{2em}{0ex}}+\sum _{k=1}^{\mathrm{\infty}}\frac{{B}_{2k+2}}{2k(2k+1)(2k+2){z}^{2k}}\end{array}$$ Originally the term $z\mathrm{Ln}\mathrm{\Gamma}\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma}\left(z+1\right)$.
Reported 20130801 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 $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime}$). The correct entries are ${k}^{1}\mathrm{ns}z$ and $k\mathrm{sn}z$. 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 $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.
$u$ $z+K$ $z+K+\mathrm{i}{K}^{\prime}$ $z+\mathrm{i}{K}^{\prime}$ $z+2K$ $z+2K+2\mathrm{i}{K}^{\prime}$ $z+2\mathrm{i}{K}^{\prime}$ $\mathrm{\vdots}$ $\mathrm{cd}u$ $\mathrm{sn}z$ ${k}^{1}\mathrm{ns}z$ ${k}^{1}\mathrm{dc}z$ $\mathrm{cd}z$ $\mathrm{cd}z$ $\mathrm{cd}z$ $\mathrm{\vdots}$ $\mathrm{dc}u$ $\mathrm{ns}z$ $k\mathrm{sn}z$ $k\mathrm{cd}z$ $\mathrm{dc}z$ $\mathrm{dc}z$ $\mathrm{dc}z$ $\mathrm{\vdots}$ Reported 20140228 by Svante Janson.
 Table 22.5.2

The entry for $\mathrm{sn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime})$ has been corrected. The correct entry is $(1+\mathrm{i})({(1+{k}^{\prime})}^{1/2}\mathrm{i}{(1{k}^{\prime})}^{1/2})/(2{k}^{1/2})$. Originally the terms ${(1+{k}^{\prime})}^{1/2}$ and ${(1{k}^{\prime})}^{1/2}$ were given incorrectly as ${(1+k)}^{1/2}$ and ${(1k)}^{1/2}$.
Similarly, the entry for $\mathrm{dn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime})$ has been corrected. The correct entry is $(1+\mathrm{i})k^{\prime}{}^{1/2}({(1+k)}^{1/2}+i{(1k)}^{1/2})/2$. Originally the terms ${(1+k)}^{1/2}$ and ${(1k)}^{1/2}$ were given incorrectly as ${(1+{k}^{\prime})}^{1/2}$ and ${(1{k}^{\prime})}^{1/2}$
Reported 20140228 by Svante Janson.
 Equation (22.6.7)

22.6.7 $$\mathrm{dn}(2z,k)=\frac{{\mathrm{dn}}^{2}(z,k){k}^{2}{\mathrm{sn}}^{2}(z,k){\mathrm{cn}}^{2}(z,k)}{1{k}^{2}{\mathrm{sn}}^{4}(z,k)}=\frac{{\mathrm{dn}}^{4}(z,k)+{k}^{2}k^{\prime}{}^{2}{\mathrm{sn}}^{4}(z,k)}{1{k}^{2}{\mathrm{sn}}^{4}(z,k)}$$ Originally the term ${k}^{2}{\mathrm{sn}}^{2}(z,k){\mathrm{cn}}^{2}(z,k)$ was given incorrectly as ${k}^{2}{\mathrm{sn}}^{2}(z,k){\mathrm{dn}}^{2}(z,k)$.
Reported 20140228 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 $0$ $\mathrm{3\hspace{0.17em}62880}$ $\mathrm{10\hspace{0.17em}26576}$ $\mathrm{11\hspace{0.17em}72700}$ $\mathrm{7\hspace{0.17em}23680}$ $\mathrm{2\hspace{0.17em}69325}$ $63273$ $9450$ $870$ $45$ $1$ Reported 20131125 by Svante Janson.
 Equation (31.8.5)

31.8.5 $${\mathrm{\Psi}}_{1,1}=\left({z}^{2}+(\lambda +3a+3)z+a\right)/{z}^{3}$$ Originally the first term on the right side of the equation for ${\mathrm{\Psi}}_{1,1}$ was ${z}^{3}$. The correct factor is ${z}^{2}$.
Reported 20130725 by Christopher Künstler.
 Equation (31.12.3)

31.12.3 $$\frac{{d}^{2}w}{{dz}^{2}}\left(\frac{\gamma}{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha zq}{z}w=0$$ Originally the sign in front of the second term in this equation was $+$. The correct sign is $$.
Reported 20131031 by Henryk Witek.
 Equation (34.4.2)

34.4.2 $$\left\{\begin{array}{ccc}\hfill {j}_{1}\hfill & \hfill {j}_{2}\hfill & \hfill {j}_{3}\hfill \\ \hfill {l}_{1}\hfill & \hfill {l}_{2}\hfill & \hfill {l}_{3}\hfill \end{array}\right\}=\begin{array}{l}\mathrm{\Delta}({j}_{1}{j}_{2}{j}_{3})\mathrm{\Delta}({j}_{1}{l}_{2}{l}_{3})\mathrm{\Delta}({l}_{1}{j}_{2}{l}_{3})\mathrm{\Delta}({l}_{1}{l}_{2}{j}_{3})\\ \phantom{\rule{2em}{0ex}}\times \sum _{s}\begin{array}{l}\frac{{(1)}^{s}(s+1)!}{(s{j}_{1}{j}_{2}{j}_{3})!(s{j}_{1}{l}_{2}{l}_{3})!(s{l}_{1}{j}_{2}{l}_{3})!(s{l}_{1}{l}_{2}{j}_{3})!}\\ \phantom{\rule{2em}{0ex}}\times \frac{1}{({j}_{1}+{j}_{2}+{l}_{1}+{l}_{2}s)!({j}_{2}+{j}_{3}+{l}_{2}+{l}_{3}s)!({j}_{3}+{j}_{1}+{l}_{3}+{l}_{1}s)!}\end{array}\end{array}$$ Originally the factor $\mathrm{\Delta}({j}_{1}{j}_{2}{j}_{3})\mathrm{\Delta}({j}_{1}{l}_{2}{l}_{3})\mathrm{\Delta}({l}_{1}{j}_{2}{l}_{3})\mathrm{\Delta}({l}_{1}{l}_{2}{j}_{3})$ was missing in this equation.
Reported 20121231 by Yu Lin.
 Other Changes

 –

–
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.

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Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.
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Crossreferences 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).
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Entries for the Sage computational system have been updated in the Software Index.

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

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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 20110810 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 $$. Previously, no conditions were stated.
Reported 20110323 by Stephen Bourn.
 Equation (10.20.14)

10.20.14 $${B}_{3}(0)=\frac{\mathrm{959\hspace{0.33em}71711\hspace{0.33em}84603}}{\mathrm{25\hspace{0.33em}47666\hspace{0.33em}37125\hspace{0.33em}00000}}{2}^{\frac{1}{3}}$$ Originally this coefficient was given incorrectly as ${B}_{3}(0)=\frac{\mathrm{430\hspace{0.33em}99056\hspace{0.33em}39368\hspace{0.33em}59253}}{\mathrm{5\hspace{0.33em}68167\hspace{0.33em}34399\hspace{0.33em}42500\hspace{0.33em}00000}}{2}^{\frac{1}{3}}$. The other coefficients in this equation have not been changed.
Reported 20120511 by Antony Lee.
 Equation (13.16.4)

The condition for the validity of this equation is $$. Originally it was given incorrectly as $\mathrm{\Re}(\kappa \mu )\frac{1}{2}>0$.  Subsection 14.2(ii)

Originally it was stated, incorrectly, that ${Q}_{\nu}^{\mu}\left(x\right)$ is real when $\nu ,\mu \in \mathrm{\mathbb{R}}$ and $x\in (1,\mathrm{\infty})$. This statement is true only for ${P}_{\nu}^{\mu}\left(x\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(x\right)$.
Reported 20120718 by Hans Volkmer and Howard Cohl.
 Equation (21.3.4)

21.3.4 $$\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha}+{\mathbf{m}}_{1}}{\mathit{\beta}+{\mathbf{m}}_{2}}\right]\left(\mathbf{z}\right\mathbf{\Omega})={\mathrm{e}}^{2\pi \mathrm{i}\mathit{\alpha}\cdot {\mathbf{m}}_{2}}\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha}}{\mathit{\beta}}\right]\left(\mathbf{z}\right\mathbf{\Omega})$$ Originally the vector ${\mathbf{m}}_{2}$ on the righthand side was given incorrectly as ${\mathbf{m}}_{1}$.
Reported 20120827 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 ${k}^{2}=\mathrm{i}{\kappa}^{2}$” (instead of ${k}^{2}=\mathrm{i}\kappa $). Also, the resolution of the graph in Figure 22.3.22 was improved near $\kappa =3$.
Reported 20111030 by Paul Abbott.
 Equation (23.2.4)

23.2.4 $$\mathrm{\wp}\left(z\right)=\frac{1}{{z}^{2}}+\sum _{w\in \mathbb{L}\backslash \{0\}}\left(\frac{1}{{(zw)}^{2}}\frac{1}{{w}^{2}}\right)$$ Originally the denominator ${(zw)}^{2}$ was given incorrectly as $(z{w}^{2})$.
Reported 20120216 by James D. Walker.
 Equation (24.4.26)

This equation is true only for $n>0$. Previously, $n=0$ was also allowed.
Reported 20120514 by Vladimir Yurovsky.
 Equation (26.12.26)

26.12.26 $$\mathit{pp}\left(n\right)\sim \frac{{\left(\zeta \left(3\right)\right)}^{7/36}}{{2}^{11/36}{(3\pi )}^{1/2}{n}^{25/36}}\mathrm{exp}\left(3{\left(\zeta \left(3\right)\right)}^{1/3}{\left(\frac{1}{2}n\right)}^{2/3}+{\zeta}^{\prime}\left(1\right)\right).$$ Originally this equation was given incorrectly as
$$\mathit{pp}\left(n\right)\sim {\left(\frac{\zeta \left(3\right)}{{2}^{11}{n}^{25}}\right)}^{1/36}\mathrm{exp}\left(3{\left(\frac{\zeta \left(3\right){n}^{2}}{4}\right)}^{1/3}+{\zeta}^{\prime}\left(1\right)\right).$$ Reported 20110905 by Suresh Govindarajan.
 Other Changes


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On August 24, 2012 Dr. Adri B. Olde Daalhuis was added as Mathematics Editor. This addition has been recorded at the end of the Preface.

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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), 20.7(iii), 20.7(vii), 25.11(iv), 25.18(i), 26.12(iv), 28.24, 28.34(ii), 29.20(i), 31.17(ii), 32.17, and as a general reference in Chapter 3.

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A crossreference was added in §21.2(i).
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Other minor changes were made in the bibliography and index.

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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 $${W}_{\frac{1}{4},\pm \frac{1}{4}}\left({z}^{2}\right)={\mathrm{e}}^{\frac{1}{2}{z}^{2}}\sqrt{\pi z}\mathrm{erfc}\left(z\right)$$ Originally the lefthand side was given correctly as ${W}_{\frac{1}{4},\frac{1}{4}}\left({z}^{2}\right)$; the equation is true also for ${W}_{\frac{1}{4},+\frac{1}{4}}\left({z}^{2}\right)$.
 Other Changes

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 $x\ge 0$. No conditions on $x$ were given originally.
Reported 20101018 by Andreas Kurt Richter.
 Figure 10.48.5

Originally the ordinate labels 2 and 4 in this figure were placed too high.
Reported 20101108 by Wolfgang Ehrhardt.
 Equation (14.19.2)

14.19.2 $${P}_{\nu \frac{1}{2}}^{\mu}\left(\mathrm{cosh}\xi \right)=\frac{\mathrm{\Gamma}\left(\frac{1}{2}\mu \right)}{{\pi}^{1/2}{\left(1{\mathrm{e}}^{2\xi}\right)}^{\mu}{\mathrm{e}}^{(\nu +(1/2))\xi}}\mathbf{F}(\frac{1}{2}\mu ,\frac{1}{2}+\nu \mu ;12\mu ;1{\mathrm{e}}^{2\xi}),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots}$. Originally the argument to $\mathbf{F}$ in this equation was incorrect (${\mathrm{e}}^{2\xi}$, rather than $1{\mathrm{e}}^{2\xi}$), and the condition on $\mu $ was too weak ($\mu \ne \frac{1}{2}$, rather than $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots}$). Also, the factor multiplying $\mathbf{F}$ was rewritten to clarify the poles; originally it was $\frac{\mathrm{\Gamma}\left(12\mu \right){2}^{2\mu}}{\mathrm{\Gamma}\left(1\mu \right){\left(1{\mathrm{e}}^{2\xi}\right)}^{\mu}{\mathrm{e}}^{(\nu +(1/2))\xi}}$.
Reported 20101102 by Alvaro Valenzuela.
 Equation (17.13.3)

17.13.3 $${\int}_{0}^{\mathrm{\infty}}{t}^{\alpha 1}\frac{{(t{q}^{\alpha +\beta};q)}_{\mathrm{\infty}}}{{(t;q)}_{\mathrm{\infty}}}dt=\frac{\mathrm{\Gamma}\left(\alpha \right)\mathrm{\Gamma}\left(1\alpha \right){\mathrm{\Gamma}}_{q}\left(\beta \right)}{{\mathrm{\Gamma}}_{q}\left(1\alpha \right){\mathrm{\Gamma}}_{q}\left(\alpha +\beta \right)},$$ Originally the differential was identified incorrectly as ${d}_{q}t$; the correct differential is $dt$.
Reported 20110408.
 Table 18.9.1

The coefficient ${A}_{n}$ for ${C}_{n}^{(\lambda )}\left(x\right)$ in the first row of this table originally omitted the parentheses and was given as $\frac{2n+\lambda}{n+1}$, instead of $\frac{2(n+\lambda )}{n+1}$.
${p}_{n}(x)$ ${A}_{n}$ ${B}_{n}$ ${C}_{n}$ ${C}_{n}^{(\lambda )}\left(x\right)$ $\frac{2(n+\lambda )}{n+1}$ $0$ $\frac{n+2\lambda 1}{n+1}$ ⋮ Reported 20100916 by Kendall Atkinson.
 Subsection 19.16(iii)

Originally it was implied that ${R}_{C}(x,y)$ is an elliptic integral. It was clarified that ${R}_{a}(\mathbf{b};\mathbf{z})$ is an elliptic integral iff the stated conditions hold; originally these conditions were stated as sufficient but not necessary. In particular, ${R}_{C}(x,y)$ does not satisfy these conditions.
Reported 20101123.
 Table 22.5.4

Originally the limiting form for $\mathrm{sc}(z,k)$ in the last line of this table was incorrect ($\mathrm{cosh}z$, instead of $\mathrm{sinh}z$).
$\mathrm{sn}(z,k)$$\to $ $\mathrm{tanh}z$ $\mathrm{cd}(z,k)$$\to $ $1$ $\mathrm{dc}(z,k)$$\to $ $1$ $\mathrm{ns}(z,k)$$\to $ $\mathrm{coth}z$ $\mathrm{cn}(z,k)$$\to $ $\mathrm{sech}z$ $\mathrm{sd}(z,k)$$\to $ $\mathrm{sinh}z$ $\mathrm{nc}(z,k)$$\to $ $\mathrm{cosh}z$ $\mathrm{ds}(z,k)$$\to $ $\mathrm{csch}z$ $\mathrm{dn}(z,k)$$\to $ $\mathrm{sech}z$ $\mathrm{nd}(z,k)$$\to $ $\mathrm{cosh}z$ $\mathrm{sc}(z,k)$$\to $ $\mathrm{sinh}z$ $\mathrm{cs}(z,k)$$\to $ $\mathrm{csch}z$ Reported 20101123.
 Equation (22.16.14)

22.16.14 $$\mathcal{E}(x,k)={\int}_{0}^{\mathrm{sn}(x,k)}\sqrt{\frac{1{k}^{2}{t}^{2}}{1{t}^{2}}}dt$$ Originally this equation appeared with the upper limit of integration as $x$, rather than $\mathrm{sn}(x,k)$.
Reported 20100708 by Charles Karney.
 Equation (26.7.6)

26.7.6 $$B\left(n+1\right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B\left(k\right)$$ Originally this equation appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$.
Reported 20101107 by Layne Watson.
 Equation (36.10.14)

36.10.14 $$3\left(\frac{{\partial}^{2}{\mathrm{\Psi}}^{(\mathrm{E})}}{{\partial x}^{2}}\frac{{\partial}^{2}{\mathrm{\Psi}}^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi}}^{(\mathrm{E})}}{\partial x}x{\mathrm{\Psi}}^{(\mathrm{E})}=0$$ Originally this equation appeared with $\frac{\partial {\mathrm{\Psi}}^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial {\mathrm{\Psi}}^{(\mathrm{E})}}{\partial x}$.
Reported 20100402.
 Other Changes


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The definition of the notation $F({z}_{0}{\mathrm{e}}^{2k\pi \mathrm{i}})$ was added in Common Notations and Definitions.
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The general references for each chapter were inserted under the isymbol on the chapter title pages. Originally these appeared only in the References sections of the individual chapters in the Handbook.

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The definition of ${R}_{C}(x,y)$ was revised in Notations.

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Additions and revisions were made in the Cross Index for Computing Special Functions

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Version 1.0.0 (May 7, 2010)
The Handbook of Mathematical Functions was published, and the Digital Library of Mathematical Functions was released.