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 1.0.13 (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 ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$. The new equations are:

25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$

$s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$.

Reported 2016-05-08 by Clemens Heuberger.

25.11.19 $${\zeta }^{\prime }(s,a)=\begin{array}{l}-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx\\ -\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,\end{array}$$

$\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

Reported 2016-06-27 by Gergő Nemes.

25.11.20 $${(-1)}^k{\zeta }^{(k)}(s,a)=\begin{array}{l}\begin{array}{l}\begin{array}{l}\frac{{(\ln a)}^k}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum _{r=0}^{k-1}\frac{{(\ln a)}^r}{r!{(s-1)}^{k-r+1}}\\ -\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^k}{{(x+a)}^{s+2}}dx\end{array}\\ +\frac{k(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-1}}{{(x+a)}^{s+2}}dx\end{array}\\ -\frac{k(k-1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-2}}{{(x+a)}^{s+2}}dx,\end{array}$$

$\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

Reported 2016-06-27 by Gergő Nemes.

Other Changes

• The symbol $\sim$ 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 $\sim$ that was used originally is incorrect and has been replaced with $\approx$, together with a slight change of wording.

• Originally (13.9.16) was expressed in term of asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right hand side, $\sim$ 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 $\overline{U}(-c,x)$. 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 ${}_r\varphi _r$ Functions to Further Transformations of ${}_{r\mathrm{+}\mathrm{1}}\varphi _r$ 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.

Reported 2015-11-12 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 2014-11-05 by Gergő Nemes.

Equation (10.32.13)

Originally the constraint, , was incorrectly written as, .

Reported 2015-05-20 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 2014-11-05 by Gergő Nemes.

Equation (23.18.7)

23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$

$c>0$.

Originally the sum ${\sum }_{r=1}^{c-1}$ 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 2015-10-05 by Howard Cohl and Tanay Wakhare.

Equations (28.28.21) and (28.28.22)

28.28.21 $$\begin{array}{l}\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\cos \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,h^2)dt\\ ={(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Mc}}_{2m+1}^{(j)}(z,h),\end{array}$$

28.28.22 $$\begin{array}{l}\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{se}}_{2m+1}(t,h^2)dt\\ ={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),\end{array}$$

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 2015-05-20 by Ruslan Kabasayev

Other Changes

• In §1.2(i), a sentence was added after (1.2.1) to refer to (1.2.6) as the definition of the binomial coefficient $\left(\genfrac{}{}{0pt}{}{z}{k}\right)$ when $z$ is complex. As a notational clarification, wherever $n$ appeared originally in (1.2.6)–(1.2.9), it was replaced by $z$.

• It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$ given in (5.11.8) is valid for $h(\in ℂ)$; 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 right-hand 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).

• Clarifications, typographic corrections, added or modified sentences appear in §§1.2(i), 1.10(i), 4.6(ii), 5.11(i), (11.11.1), (11.11.9), (21.5.7), and (27.14.7).

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 $p(\ne 0)$ 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:

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

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

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, 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 2014-10-31 by Masataka Urago.

Equation (9.10.18)

9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt.$$

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

Reported 2015-03-20 by Walter Gautschi.

Equation (9.10.19)

9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt.$$

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

Reported 2015-03-20 by Walter Gautschi.

Equation (10.17.14)

10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right)$$

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 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 right-hand side of this equation was written incorrectly as $-\frac{549}{28000}{a}^8$.

Reported 2015-03-16 by Svante Janson.

Equation (13.2.7)

13.2.7 $$U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0pt}{}{m}{s}\right){\left(b+s\right)}_{m-s}{(-z)}^s$$

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 2015-02-10 by Adri Olde Daalhuis.

Equation (13.2.8)

13.2.8 $$U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)=z^{-a}\sum _{s=0}^n\left(\genfrac{}{}{0pt}{}{n}{s}\right){\left(a\right)}_s{z}^{-s}$$

The equality $U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)$ 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 $$U(-m,n+1,z)={(-1)}^m{\left(n+1\right)}_{m}M(-m,n+1,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0pt}{}{m}{s}\right){\left(n+s+1\right)}_{m-s}{(-z)}^s$$

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 2015-02-10 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 2014-10-03 by Roderick Wong.

Equation (34.7.4)

34.7.4

Originally the third $\mathit{3}j$ symbol in the summation was written incorrectly as

Reported 2015-01-19 by Yan-Rui 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 left-hand sides have all been changed to $\mathrm{Ln}\mathrm{\Gamma }(\cdot )$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }(\cdot )$ was used, where $\ln$ is the principal branch of the logarithm. These changes were recommended by Philippe Spindel on 2015-02-06.

• The notation used for the $q$-Appell functions in Section 17.1 and Equations (17.4.5), (17.4.6),(17.4.7), (17.4.8), (17.11.1), (17.11.2) and (17.11.3) was updated to explicitly include the argument $q$, as used in Gasper and Rahman (2004).

• 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 $\ln$ from $\log$ 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 2014-05-21 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 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 . 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 2014-05-02 by Svante Janson.

Equation (34.3.7)

34.3.7

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 2014-06-12 by James Zibin.

Other Changes

• Pochhammer symbols have been introduced in Equations (7.12.1), (7.12.2), (7.12.3), (7.12.4), (7.12.5), (25.5.7), (25.8.3), (25.11.10), (25.11.28), and (25.11.43) to make the notation more concise.

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

• A cross-reference has been added in §15.9(iv).

• Equations (22.19.6), (22.19.7), (22.19.8), and (22.19.9) have been rewritten with the modulus (second argument) of the Jacobian elliptic function defined explicitly in the preceding line of text.

• Bibliographic citations have been added in §§4.13, 4.48(iv), 6.21(ii), 8.28(ii), 9.16, 10.77(viii), 12.21(ii), 14.28(ii), 14.34(ii), 16.4(ii) and 16.13.

• 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 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right))$$

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 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 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 $\mathrm{e}$, imaginary $\mathrm{i}$ 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 $J_0\left(t\right)$ 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 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 $(\pm \sqrt{\frac{3}{5}}h,\pm \sqrt{\frac{3}{5}}h)$. 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 2014-01-13 by Stanley Oleszczuk.

Equation (4.21.1)

4.21.1 $$\sin u\pm \cos u=\sqrt{2}\sin \left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\cos \left(u\mp \frac{1}{4}\pi \right)$$

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)

4.23.34 $$\arcsin z=\arcsin \beta +\mathrm{i}\mathrm{sign}\left(y\right)\ln \left(\alpha +{({\alpha }^2-1)}^{1/2}\right)$$

and

4.23.35 $$\arccos z=\arccos \beta -\mathrm{i}\mathrm{sign}\left(y\right)\ln \left(\alpha +{({\alpha }^2-1)}^{1/2}\right)$$

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 2013-07-01 by Volker Thürey.

Equation (5.17.5)

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \begin{array}{l}\frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\mathrm{Ln}z-\ln A\\ +\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}\end{array}$$

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

Reported 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 $\mathrm{cd}u$ 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.

	$u$
	$z+K$	$z+K+\mathrm{i}{K}^{\prime }$	$z+\mathrm{i}{K}^{\prime }$	$z+2K$	$z+2K+2\mathrm{i}{K}^{\prime }$	$z+2\mathrm{i}{K}^{\prime }$
$\mathrm{⋮}$
$\mathrm{cd}u$	$-\mathrm{sn}z$	$-k^{-1}\mathrm{ns}z$	$k^{-1}\mathrm{dc}z$	$-\mathrm{cd}z$	$-\mathrm{cd}z$	$\mathrm{cd}z$
$\mathrm{⋮}$
$\mathrm{dc}u$	$-\mathrm{ns}z$	$-k\mathrm{sn}z$	$k\mathrm{cd}z$	$-\mathrm{dc}z$	$-\mathrm{dc}z$	$\mathrm{dc}z$
$\mathrm{⋮}$

Reported 2014-02-28 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 $(1+\mathrm{i})({(1+k^{\prime })}^{1/2}-\mathrm{i}{(1-k^{\prime })}^{1/2})/(2k^{1/2})$. 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 $\mathrm{dn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$ has been corrected. The correct entry is $(-1+\mathrm{i})k^{\prime }{}^{1/2}({(1+k)}^{1/2}+i{(1-k)}^{1/2})/2$. 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)

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 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	$0$	$-\mathrm{3\hspace{0.17em}62880}$	$\mathrm{10\hspace{0.17em}26576}$	$-\mathrm{11\hspace{0.17em}72700}$	$\mathrm{7\hspace{0.17em}23680}$	$-\mathrm{2\hspace{0.17em}69325}$	$63273$	$-9450$	$870$	$-45$	$1$

Reported 2013-11-25 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 2013-07-25 by Christopher Künstler.

Equation (31.12.3)

31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0$$

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)

34.4.2

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 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 $\arctan x$.

• 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.

• Bibliographic citations have been added in §§4.13, 5.6(i), 5.11(iii), 7.25(iii), 8.13(i), 10.37, 12.18, 14.11, 15.12(ii), 16.6, 18.16(ii), 18.16(iv), 18.24, 18.27(iv), 18.27(v),18.28(i), 24.13(i), 28.36(iii).

• 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 . Previously, no conditions were stated.

Reported 2011-03-23 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 2012-05-11 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 $\nu ,\mu \in \mathrm{ℝ}$ 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 2012-07-18 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 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 $k^2=\mathrm{i}{\kappa }^2$” (instead of $k^2=\mathrm{i}\kappa$). Also, the resolution of the graph in Figure 22.3.22 was improved near $\kappa =3$.

Reported 2011-10-30 by Paul Abbott.

Equation (23.2.4)

23.2.4 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{w\in 𝕃\backslash \{0\}}\left(\frac{1}{{(z-w)}^2}-\frac{1}{w^2}\right)$$

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)

26.12.26 $$\mathit{pp}\left(n\right)\sim \frac{{\left(\zeta \left(3\right)\right)}^{7/36}}{2^{11/36}{(3\pi )}^{1/2}{n}^{25/36}}\exp \left(3{\left(\zeta \left(3\right)\right)}^{1/3}{\left(\frac{1}{2}n\right)}^{2/3}+{\zeta }^{\prime }\left(-1\right)\right).$$

Originally this equation was given incorrectly as

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

Reported 2011-09-05 by Suresh Govindarajan.

Other Changes

• 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.

• 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.

• A cross-reference was added in §21.2(i).

• 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, 20.7(iv), 21.2(i), 30.13(i), 30.14(i), 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)

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 left-hand 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

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 $x\ge 0$. 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)

14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}\mathbf{F}(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$

$\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$.

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 $\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}$.

Reported 2010-11-02 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 2011-04-08.

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 2010-09-16 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 2010-11-23.

Table 22.5.4

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

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

Reported 2010-11-23.

Equation (22.16.14)

22.16.14 $$ℰ(x,k)={\int }_0^{\mathrm{sn}(x,k)}\sqrt{\frac{1-k^2{t}^2}{1-t^2}}dt$$

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

Reported 2010-07-08 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 2010-11-07 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 2010-04-02.

Other Changes

• The definition of the notation $F(z_0{\mathrm{e}}^{2k\pi \mathrm{i}})$ 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 $R_C(x,y)$ 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.