Sinc function

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11: Mathematical Introduction

… ►The mathematical project team has endeavored to take into account the hundreds of research papers and numerous books on special functions that have appeared since 1964. …

12: 31.11 Expansions in Series of Hypergeometric Functions

… ►The series of Type I (§31.11(iii)) are useful since they represent the functions in large domains. …

13: Foreword

… ►In 1964 the National Institute of Standards and Technology published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. …The provision of standard reference data of this type is a core function of NIST. ►Much has changed in the years since A&S was published. …The new printed volume, the NIST Handbook of Mathematical Functions, serves a similar function as the original A&S, though it is heavily updated and extended. The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. …

14: 25.11 Hurwitz Zeta Function

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25.11.30 $$\zeta (s,a)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{{\mathrm{e}}^{az}{z}^{s-1}}{1-{\mathrm{e}}^z}dz,$$ $s\ne 1$, $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

contour integral, integral representation

Source:

Erdélyi et al. (1953a, (1.10.5), p. 25); with $z=-t$

Proof sketch:

This contour integral uses Hankel’s contour Figure 5.9.1. Assume $\mathrm{\Re }s>1$, collapse the integration path onto the negative real axis, apply (25.11.25), followed by analytic continuation for all $s\ne 1$, $\mathrm{\Re }a>0$, since the integrand is analytic in $z$, the convergence at the ends of the path is exponential for all $s\ne -1$ and the Hankel contour can be kept well clear of the singularity at the origin in the $z$-plane.

Referenced by:

§25.11(i)

Permalink:

http://dlmf.nist.gov/25.11.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vii), §25.11 and Ch.25

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25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $k$: nonnegative integer

Referenced by:

§25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.11.E36

Removal (effective with 1.1.4):

This equation has been removed because it is just (25.15.1) combined with (25.15.3).

Clarification (effective with 1.0.23):

The Dirichlet $L$-function was inserted on the left-hand side. The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

See also:

Annotations for §25.11(x), §25.11 and Ch.25

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25.11.37 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}\zeta (nk,a)=-n\ln \mathrm{\Gamma }\left(a\right)+\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(a-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$, $\mathrm{\Re }a\ge 1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

infinite series, series representation

Source:

Adamchik and Srivastava (1998, (2.12), p. 136); with the choice $\mathrm{ph}\left(-1\right)=\pi$, which is reasonable since the gamma function is single valued

Permalink:

http://dlmf.nist.gov/25.11.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xi), §25.11 and Ch.25

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15: 20.1 Special Notation

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$m$, $n$	integers.
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$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
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16: 32.10 Special Function Solutions

… ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime },C)=0$, with $C$ a constant. …

17: Errata

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The constraints in (14.5.3), (14.5.4) on $\nu +\mu$ have been corrected to exclude all negative integers since the Ferrers function of the second kind is not defined for these values.

Reported by Hans Volkmer on 2021-06-02

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18: 14.15 Uniform Asymptotic Approximations

§14.15 Uniform Asymptotic Approximations

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§14.15(i) Large $\mu$, Fixed $\nu$

… ►(The inverse hyperbolic functions again take their principal values.) ►Since (14.15.30) holds for negative $x$, corresponding approximations for ${𝖰}_{\nu }^{\mp \mu }\left(x\right)$, uniformly valid in the interval , can be obtained from (14.9.9) and (14.9.10). … ►

19: 28.2 Definitions and Basic Properties

… ►Since (28.2.1) has no finite singularities its solutions are entire functions of $z$. …

20: 31.9 Orthogonality

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§31.9(i) Single Orthogonality

… ►The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … ►The right-hand side may be evaluated at any convenient value, or limiting value, of $\zeta$ in $(0,1)$ since it is independent of $\zeta$. ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

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