analytic continuation

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11: 8.15 Sums

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8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

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12: 33.23 Methods of Computation

… ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …

13: 36.12 Uniform Approximation of Integrals

… ►If , then we may evaluate the complex conjugate of $I$ for real values of $𝐲$ and $g$, and obtain $I$ by conjugation and analytic continuation. … ►The square roots are real and positive when $𝐲$ is such that all the critical points are real, and are defined by analytic continuation elsewhere. … ►Also, ${\mathrm{\Delta }}^{1/4}/\sqrt{f_{+}^{\prime \prime }}$ and ${\mathrm{\Delta }}^{1/4}/\sqrt{-f_{-}^{\prime \prime }}$ are chosen to be positive real when $y$ is such that both critical points are real, and by analytic continuation otherwise. …

14: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector ; compare §16.2(iii). …

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

16: 8.19 Generalized Exponential Integral

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§8.19(viii) Analytic Continuation

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17: 11.4 Basic Properties

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§11.4(iii) Analytic Continuation

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18: 5.2 Definitions

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

19: 8.21 Generalized Sine and Cosine Integrals

… ►Elsewhere in the sector $|\mathrm{ph}z|\le \pi$ the principal values are defined by analytic continuation from $\mathrm{ph}z=0$; compare §4.2(i). …

20: 30.6 Functions of Complex Argument

… ►of (30.2.1) with $\mu =m$ and $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$ are real when $z\in (1,\mathrm{\infty })$, and their principal values (§4.2(i)) are obtained by analytic continuation to $ℂ\setminus (-\mathrm{\infty },1]$. …