analytic continuation onto higher Riemann sheets

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11: 16.2 Definition and Analytic Properties

§16.2 Definition and Analytic Properties

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§16.2(ii) Case $p\le q$

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§16.2(iii) Case $p=q+1$

… ►If none of the $a_j$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to $z$. … ►

§16.2(iv) Case $p>q+1$

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12: 10.34 Analytic Continuation

§10.34 Analytic Continuation

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13: 14.24 Analytic Continuation

§14.24 Analytic Continuation

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14: 5.2 Definitions

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5.2.1 $$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $\mathrm{\Re }z>0$.

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Defines:

$\mathrm{\Gamma }\left(z\right)$: gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

6.1.1

Referenced by:

§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)

Permalink:

http://dlmf.nist.gov/5.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

►When $\mathrm{\Re }z\le 0$, $\mathrm{\Gamma }\left(z\right)$ is defined by analytic continuation. … ►

5.2.2 $$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$ $z\ne 0,-1,-2,\mathrm{\dots }$.

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Defines:

$\psi \left(z\right)$: psi (or digamma) function

Symbols:

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

A&S Ref:

6.3.1

Permalink:

http://dlmf.nist.gov/5.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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15: 15.2 Definitions and Analytical Properties

§15.2 Definitions and Analytical Properties

… ►on the disk , and by analytic continuation elsewhere. … ►again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. … ►

§15.2(ii) Analytic Properties

… ►The right-hand side can be seen as an analytical continuation for the left-hand side when $a$ approaches $-m$. …

16: 2.4 Contour Integrals

… ►If $q(t)$ is analytic in a sector containing $\mathrm{ph}t=0$, then the region of validity may be increased by rotation of the integration paths. … ►is continuous in $\mathrm{\Re }z\ge c$ and analytic in $\mathrm{\Re }z>c$, and by inversion (§1.14(iii)) … ►Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion … ►Assume that $p(t)$ and $q(t)$ are analytic on an open domain $𝐓$ that contains $𝒫$, with the possible exceptions of $t=a$ and $t=b$. … ►in which $z$ is a large real or complex parameter, $p(\alpha ,t)$ and $q(\alpha ,t)$ are analytic functions of $t$ and continuous in $t$ and a second parameter $\alpha$. …

17: 10.11 Analytic Continuation

§10.11 Analytic Continuation

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18: 8.19 Generalized Exponential Integral

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§8.19(viii) Analytic Continuation

… ►For higher-order generalized exponential integrals see Meijer and Baken (1987) and Milgram (1985).

19: 25.2 Definition and Expansions

§25.2 Definition and Expansions

… ►Elsewhere $\zeta \left(s\right)$ is defined by analytic continuation. … ►

§25.2(ii) Other Infinite Series

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§25.2(iii) Representations by the Euler–Maclaurin Formula

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§25.2(iv) Infinite Products

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20: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

… ► $\zeta (s,a)$ has a meromorphic continuation in the $s$-plane, its only singularity in $ℂ$ being a simple pole at $s=1$ with residue $1$. As a function of $a$, with $s$ ($\ne 1$) fixed, $\zeta (s,a)$ is analytic in the half-plane $\mathrm{\Re }a>0$. The Riemann zeta function is a special case: … ►

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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