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11: 25.14 Lerch’s Transcendent

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25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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12: 31.7 Relations to Other Functions

… ►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. …

13: 32.9 Other Elementary Solutions

… ►Then ${\text{P}}_{\text{III}}$ has algebraic solutions iff …

14: 18.2 General Orthogonal Polynomials

… ►The OP’s are orthonormal iff $C_n=A_n/A_{n-1}$ ($n\ge 1$) and $h_0=1$. The OP’s are monic iff $A_n=1$ ($n\ge 0$) and $k_0=1$. … ►The OP’s are orthonormal iff $c_n=a_{n-1}$ ($n\ge 1$) and $h_0=1$. The OP’s are monic iff $a_n=1$ ($n\ge 0$) and $k_0=1$. … ►This matrix is symmetric iff $c_n=a_{n-1}$ ($n\ge 1$). …

15: 1.10 Functions of a Complex Variable

… ►Equalities hold iff $f(z)=Az$, where $A$ is a constant such that $\left|A\right|=M/R$. … ►The product ${\prod }_{n=1}^{\mathrm{\infty }}(1+a_n)$, with $a_n\ne -1$ for all $n$, converges iff ${\sum }_{n=1}^{\mathrm{\infty }}\ln \left(1+a_n\right)$ converges; and it converges absolutely iff ${\sum }_{n=1}^{\mathrm{\infty }}\left|a_n\right|$ converges. …

16: 19.16 Definitions

… ► $R_{-a}(𝐛;𝐳)$ is an elliptic integral iff the $z$’s are distinct and exactly four of the parameters $a,a^{\prime },b_1,\mathrm{\dots },b_n$ are half-odd-integers, the rest are integers, and none of $a$, $a^{\prime }$, $a+a^{\prime }$ is zero or a negative integer. …

17: Mathematical Introduction

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$ℂ$	complex plane (excluding infinity).
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iff	if and only if.
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18: 23.5 Special Lattices

… ►The Weierstrass functions take real values on the real axis iff the lattice is fixed under complex conjugation: $𝕃=\overline{𝕃}$; equivalently, when $g_2,g_3\in ℝ$. …

19: 1.4 Calculus of One Variable

… ►If $f(x)$ is twice differentiable, then $f(x)$ is convex iff $f^{\prime \prime }(x)\ge 0$ on $(a,b)$. A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …

20: 1.2 Elementary Algebra

… ►Multiplication of an $m\times n$ matrix $𝐀$ and an $m^{\prime }\times n^{\prime }$ matrix $𝐁$, giving the $m\times n^{\prime }$ matrix $𝐂=𝐀𝐁$ is defined iff $n=m^{\prime }$. … ►the equality holding iff $𝐯$ is a scalar (real or complex) multiple of $𝐮$. …