contour integrals

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21: Bibliography R

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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

External Links:

ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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22: Bibliography F

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Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.

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External Links:

ISSN 1058-6458, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

(25.16.13), §25.16(ii), §25.16(ii), §25.16.

Encodings:

BibTeX

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23: 14.12 Integral Representations

… ►For further integral representations see Erdélyi et al. (1953a, pp. 158–159) and Magnus et al. (1966, pp. 184–190), and for contour integrals and other representations see §14.25.

24: 16.2 Definition and Analytic Properties

… ►See §16.5 for the definition of ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ as a contour integral when $p>q+1$ and none of the $a_k$ is a nonpositive integer. …

25: 9.13 Generalized Airy Functions

… ►Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … ►Further properties of these functions, and also of similar contour integrals containing an additional factor ${(\ln t)}^q$, $q=1,2,\mathrm{\dots }$, in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). …

26: 2.10 Sums and Sequences

… ►The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. … ►

2.10.20 $$\sum _{j=0}^{n-1}\frac{x^j}{{(j!)}^3}=\frac{1}{2\mathrm{i}}{\int }_{𝒞}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}\cot \left(\pi t\right)dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral and $𝒞$: contour

Permalink:

http://dlmf.nist.gov/2.10.E20

Encodings:

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See also:

Annotations for §2.10(iii), §2.10(iii), §2.10 and Ch.2

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2.10.22 $$\sum _{j=0}^{n-1}\frac{x^j}{{(j!)}^3}={\int }_{-1/2}^{n-(1/2)}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}dt-{\int }_{{𝒞}_1}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}\frac{dt}{{\mathrm{e}}^{-2\pi \mathrm{i}t}-1}+{\int }_{{𝒞}_2}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}\frac{dt}{{\mathrm{e}}^{2\pi \mathrm{i}t}-1},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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 $𝒞$: contour

Permalink:

http://dlmf.nist.gov/2.10.E22

Encodings:

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See also:

Annotations for §2.10(iii), §2.10(iii), §2.10 and Ch.2

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2.10.26 $$f_n=\frac{1}{2\pi \mathrm{i}}{\int }_{𝒞}\frac{f(z)}{z^{n+1}}dz,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $f(x)$: analytic function, $f_n$: coefficients and $𝒞$: simple closed contour

Permalink:

http://dlmf.nist.gov/2.10.E26

Encodings:

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See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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27: 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:

contourintegral, 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

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See also:

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

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28: Bibliography S

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Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45 (2), pp. 558–571.

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External Links:

ISSN 0036-1429, Document, MathReview (Amparo Gil), ZentralBlatt

Referenced by:

§5.21, §5.23(iii).

Encodings:

BibTeX

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29: 5.12 Beta Function

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Figure 5.12.1: $t$-plane. Contour for first loop integral for the beta function. Magnify

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Figure 5.12.2: $t$-plane. Contour for second loop integral for the beta function. Magnify

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Figure 5.12.3: $t$-plane. Contour for Pochhammer’s integral. Magnify

30: 31.9 Orthogonality

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31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

Permalink:

http://dlmf.nist.gov/31.9.E2

Encodings:

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See also:

Annotations for §31.9(i), §31.9 and Ch.31

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