based on Sinc functions

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1: 3.3 Interpolation

… ►For interpolation of a bounded function $f$ on $ℝ$ the cardinal function of $f$ is defined by …

2: Bibliography S

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

Encodings:

BibTeX

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3: 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

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4: 3.4 Differentiation

… ►For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). … ►

§3.4(ii) Analytic Functions

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3.4.18 $$\frac{1}{k!}{f}^{(k)}(x_0)=\frac{1}{2\pi r^k}{\int }_0^{2\pi }f(x_0+r{\mathrm{e}}^{\mathrm{i}\theta }){\mathrm{e}}^{-\mathrm{i}k\theta }d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral and $r$: radius

Referenced by:

§3.4(ii)

Permalink:

http://dlmf.nist.gov/3.4.E18

Encodings:

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See also:

Annotations for §3.4(ii), §3.4 and Ch.3

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3.4.19 $$\frac{1}{k!}=\frac{1}{2\pi r^k}{\int }_0^{2\pi }{\mathrm{e}}^{r\cos \theta }\cos \left(r\sin \theta -k\theta \right)d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function and $r$: radius

Permalink:

http://dlmf.nist.gov/3.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(ii), §3.4(ii), §3.4 and Ch.3

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5: 33.11 Asymptotic Expansions for Large $\rho$

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient and $b$: coefficient

A&S Ref:

14.5.9

Referenced by:

Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)

Permalink:

http://dlmf.nist.gov/33.11.E1

Encodings:

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Errata (effective with 1.0.22):

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported 2019-01-29 by Gergő Nemes

Errata (effective with 1.0.19):

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.11 and Ch.33

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6: 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), §25.11(vii), Erratum (V1.2.1) for Figure 5.9.1

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

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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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7: 4.45 Methods of Computation

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Logarithms

… ►and since $|y|\le \frac{1}{2}\ln 10=1.15\mathrm{\dots }$, ${\mathrm{e}}^y$ can be computed straightforwardly from (4.2.19). ►

Trigonometric Functions

… ►and since $|\theta |\le \frac{1}{2}\pi =1.57\mathrm{\dots }$, $\sin \theta$ and $\cos \theta$ can be computed straightforwardly from (4.19.1) and (4.19.2). … ►

Inverse Trigonometric Functions

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8: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

… ► … ►When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … ►

§25.15(ii) Zeros

►Since $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L(s,\chi )$ for (the so-called trivial zeros) are as follows: …

9: 1.16 Distributions

… ► … ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►Since ${\delta }_{x_0}$ is the Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ corresponding to $\alpha (x)=H\left(x-x_0\right)$ (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of $\alpha$. … ►Since $\sqrt{2\pi }ℱ\left(\delta \right)=1$, we have …Since the quantity on the extreme right of (1.16.41) is equal to $\sqrt{2\pi }⟨\delta ,\varphi ⟩$, as distributions, the result in this equation can be stated as …

10: 2.3 Integrals of a Real Variable

… ►For the function $\mathrm{\Gamma }$ see §5.2(i). ►This result is probably the most frequently used method for deriving asymptotic expansions of special functions. Since $q(t)$ need not be continuous (as long as the integral converges), the case of a finite integration range is included. … ►Another extension is to more general factors than the exponential function. … ►

If $p(b)=\mathrm{\infty }$, then $P_0(b)=0$ and each of the integrals

2.3.22 $$\int {\mathrm{e}}^{\mathrm{i}xp(t)}{P}_s(t)p^{\prime }(t)dt,$$ $s=0,1,2,\mathrm{\dots }$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $p(t)$: real function and $P_s(t)$: function

Permalink:

http://dlmf.nist.gov/2.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iv), §2.3 and Ch.2

converges at $t=b$ uniformly for all sufficiently large $x$.

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