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21: Errata

… ►

A sentence was added in §8.18(ii) to refer to Nemes and Olde Daalhuis (2016). Originally §8.11(iii) was applicable for real variables $a$ and $x=\lambda a$. It has been extended to allow for complex variables $a$ and $z=\lambda a$ (and we have replaced $x$ with $z$ in the subsection heading and in Equations (8.11.6) and (8.11.7)). Also, we have added two paragraphs after (8.11.9) to replace the original paragraph that appeared there. Furthermore, the interval of validity of (8.11.6) was increased from to the sector , and the interval of validity of (8.11.7) was increased from $\lambda >1$ to the sector $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$. A paragraph with reference to Nemes (2016) has been added in §8.11(v), and the sector of validity for (8.11.12) was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$. Two new Subsections 13.6(vii), 13.18(vi), both entitled Coulomb Functions, were added to note the relationship of the Kummer and Whittaker functions to various forms of the Coulomb functions. A sentence was added in both §13.10(vi) and §13.23(v) noting that certain generalized orthogonality can be expressed in terms of Kummer functions.

…

22: 31.7 Relations to Other Functions

… ►Below are three such reductions with three and two parameters. … ►

31.7.2 $$H\mathrm{\ell }(2,\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta -2\gamma +1;z)={}_2{}^{}F_1^{}(\frac{1}{2}\alpha ,\frac{1}{2}\beta ;\gamma ;1-{(1-z)}^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E2

Encodings:

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See also:

Annotations for §31.7(i), §31.7 and Ch.31

… ►Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. … ►With $z={\mathrm{sn}}^2(\zeta ,k)$ and …equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). …

23: 15.8 Transformations of Variable

… ►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. … ►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). … ►When the intersection of two groups in Table 15.8.1 is not empty there exist special quadratic transformations, with only one free parameter, between two hypergeometric functions in the same group. … ►This is a quadratic transformation between two cases in Group 1. … ►which is a quadratic transformation between two cases in Group 3. …

24: 6.4 Analytic Continuation

… ►

6.4.1 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\mathrm{Ln}z-\gamma ;$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E1

Encodings:

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Annotations for §6.4 and Ch.6

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6.4.2 $$E_1\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)=E_1\left(z\right)-2m\pi \mathrm{i},$$ $m\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers and $z$: complex variable

Permalink:

http://dlmf.nist.gov/6.4.E2

Encodings:

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Annotations for §6.4 and Ch.6

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6.4.4 $$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E4

Encodings:

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See also:

Annotations for §6.4 and Ch.6

►

6.4.5 $$\mathrm{Chi}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Chi}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E5

Encodings:

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See also:

Annotations for §6.4 and Ch.6

… ►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_1\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis. …

25: 31.15 Stieltjes Polynomials

… ►

31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

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See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.3 $$\sum _{j=1}^N\frac{{\gamma }_j}{t_k-a_j}+\sum _{j=1}^{n-1}\frac{1}{t_k-z_j^{\prime }}=0.$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $t_k$: zero of $V(z)$

Permalink:

http://dlmf.nist.gov/31.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

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See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►Let $S_{𝐦}(z)$ and $S_{𝐥}(z)$ be Stieltjes polynomials corresponding to two distinct multi-indices $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$ and $𝐥=({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathrm{\dots },{\mathrm{\ell }}_{N-1})$. … ►

31.15.8 $$S_{𝐦}(z_1)S_{𝐦}(z_2)\mathrm{\cdots }{S}_{𝐦}(z_{N-1}),$$ $z_j\in (a_j,a_{j+1})$,

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Symbols:

$\in$: element of, $(a,b)$: open interval, $z$: complex variable, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $S_{𝐦}(z)$: Stieltjes polynomial

Referenced by:

§31.15(iii)

Permalink:

http://dlmf.nist.gov/31.15.E8

Encodings:

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See also:

Annotations for §31.15(iii), §31.15 and Ch.31

…

26: 13.5 Continued Fractions

… ►If $a,b\in ℂ$ such that $a\ne -1,-2,-3,\mathrm{\dots }$, and $a-b\ne 0,1,2,\mathrm{\dots }$, then ►

13.5.1 $$\frac{M(a,b,z)}{M(a+1,b+1,z)}=1+\frac{u_{1}z}{1+}\frac{u_{2}z}{1+}\mathrm{\cdots },$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $z$: complex variable and $u_n$: continued fraction coefficients

Referenced by:

§13.31(ii)

Permalink:

http://dlmf.nist.gov/13.5.E1

Encodings:

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Annotations for §13.5 and Ch.13

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $ℂ$. … ►If $a,b\in ℂ$ such that $a\ne 0,-1,-2,\mathrm{\dots }$, and $b-a\ne 2,3,4,\mathrm{\dots }$, then ►

13.5.3 $$\frac{U(a,b,z)}{U(a,b-1,z)}=1+\frac{v_1/z}{1+}\frac{v_2/z}{1+}\mathrm{\cdots },$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $z$: complex variable and $v_n$: continued fraction coefficients

Permalink:

http://dlmf.nist.gov/13.5.E3

Encodings:

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See also:

Annotations for §13.5 and Ch.13

…

27: 8.15 Sums

… ►

8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

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Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

…

28: 4.13 Lambert $W$-Function

… ►

4.13.1 $$W{\mathrm{e}}^W=z.$$

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Defines:

$W\left(z\right)$: Lambert $W$-function

Symbols:

$\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§4.13

Permalink:

http://dlmf.nist.gov/4.13.E1

Encodings:

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See also:

Annotations for §4.13 and Ch.4

… ►On the $z$-interval $(-{\mathrm{e}}^{-1},0)$ there are two real solutions, one increasing and the other decreasing. … ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►

4.13.5_1 $${\left(\frac{W_0\left(z\right)}{z}\right)}^a={\mathrm{e}}^{-a{W}_0\left(z\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{a{\left(n+a\right)}^{n-1}}{n!}{\left(-z\right)}^n,$$ , $a\in ℂ$.

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $ℂ$: complex plane, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $a$: real or complex constant and $z$: complex variable

Notes:

See Corless et al. (1996, formula (2.36)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E5_1

Encodings:

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Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

…

29: 28.12 Definitions and Basic Properties

… ►Two eigenfunctions correspond to each eigenvalue $a={\lambda }_{\nu }\left(q\right)$. … ►

28.12.4 $${\mathrm{me}}_{\nu }(z,0)={\mathrm{e}}^{\mathrm{i}\nu z}.$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E4

Encodings:

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See also:

Annotations for §28.12(ii), §28.12 and Ch.28

… ►

28.12.8 $${\mathrm{me}}_{-\nu }(z,q)={\mathrm{me}}_{\nu }(-z,q),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

… ►(28.12.10) is not valid for cuts on the real axis in the $q$-plane for special complex values of $\nu$; but it remains valid for small $q$; compare §28.7. … ►These functions are real-valued for real $\nu$, real $q$, and $z=x$, whereas ${\mathrm{me}}_{\nu }(x,q)$ is complex. …

30: 16.11 Asymptotic Expansions

… ►For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows: … ►

16.11.2 $$H_{p,q}(z)=\sum _{m=1}^p\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\mathrm{\Gamma }\left(a_m+k\right)\left(\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne m\end{array}}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}-a_m-k\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-a_m-k\right)\right)z^{-a_m-k}.$$

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Defines:

$H_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.11(ii), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E2

Encodings:

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See also:

Annotations for §16.11(i), §16.11 and Ch.16

… ►It may be observed that $H_{p,q}(z)$ represents the sum of the residues of the poles of the integrand in (16.5.1) at $s=-a_j,-a_j-1,\mathrm{\dots }$, $j=1,\mathrm{\dots },p$, provided that these poles are all simple, that is, no two of the $a_j$ differ by an integer. … ►

§16.11(ii) Expansions for Large Variable

… ►

16.11.6 $$\left(\prod _{\mathrm{\ell }=1}^{q+1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=H_{q+1,q}(-z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$;

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $H_{p,q}(z)$: formal infinite series

Permalink:

http://dlmf.nist.gov/16.11.E6

Encodings:

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See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

…