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21: 12.5 Integral Representations

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§12.5(ii) Contour Integrals

… ►where the contour separates the poles of $\mathrm{\Gamma }\left(t\right)$ from those of $\mathrm{\Gamma }\left(\frac{1}{2}+a-2t\right)$. … ►where the contour separates the poles of $\mathrm{\Gamma }\left(t\right)$ from those of $\mathrm{\Gamma }\left(\frac{1}{2}-a-2t\right)$. …

22: 16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. ►Suppose first that $L$ is a contour that starts at infinity on a line parallel to the positive real axis, encircles the nonnegative integers in the negative sense, and ends at infinity on another line parallel to the positive real axis. … ►Secondly, suppose that $L$ is a contour from $-\mathrm{i}\mathrm{\infty }$ to $\mathrm{i}\mathrm{\infty }$. … ►Lastly, the restrictions on the parameters can be eased by replacing the integration paths with loop contours; see Luke (1969a, §3.6). …

23: 25.5 Integral Representations

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§25.5(iii) Contour Integrals

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25.5.20 $$\zeta \left(s\right)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}}^{-z}-1}dz,$$ $s\ne 1,2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann 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, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

contour integral, integral representation

Sources:

Apostol (1976, (6), p. 253); with $a=1$; Erdélyi et al. (1953a, (1.12.9), p. 32); with $z=-t$

A&S Ref:

23.2.4 (in slightly different form)

Referenced by:

§25.11(vii)

Permalink:

http://dlmf.nist.gov/25.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(iii), §25.5 and Ch.25

►where the integration contour is a loop around the negative real axis; it starts at $-\mathrm{\infty }$, encircles the origin once in the positive direction without enclosing any of the points $z=\pm 2\pi \mathrm{i}$, $\pm 4\pi \mathrm{i}$, …, and returns to $-\mathrm{\infty }$. … ►

25.5.21 $$\zeta \left(s\right)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}(1-2^{1-s})}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}}^{-z}+1}dz,$$ $s\ne 1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann 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, $s$: complex variable and $z$: complex variable

Keywords:

contour integral, integral representation

Sources:

Erdélyi et al. (1953a, (1.12.10), p. 32); with $z=-t$; Magnus et al. (1966, p. 21); with $z=-t$

Permalink:

http://dlmf.nist.gov/25.5.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(iii), §25.5 and Ch.25

►The contour here is any loop that encircles the origin in the positive direction not enclosing any of the points $\pm \pi \mathrm{i}$, $\pm 3\pi \mathrm{i}$, ….

24: 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. … ►

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Figure 2.10.1: $t$-plane. Contour $𝒞$. Magnify

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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:

pMML, png, TeX

See also:

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

►where $𝒞$ is a simple closed contour in the annulus that encloses $z=0$. … ►By allowing the contour in Cauchy’s formula to expand, we find that …

25: 8.6 Integral Representations

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§8.6(ii) Contour Integrals

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26: 9.17 Methods of Computation

… ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …

27: 11.5 Integral Representations

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§11.5(ii) Contour Integrals

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28: 14.25 Integral Representations

… ►For corresponding contour integrals, with less restrictions on $\mu$ and $\nu$, see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).

29: Publications

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Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

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30: 21.1 Special Notation

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$g,h$	positive integers.
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${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

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