absolutely convergent

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11: 20.5 Infinite Products and Related Results

… ►With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the $z$-plane. … ►These double products are not absolutely convergent; hence the order of the limits is important. …

12: 1.17 Integral and Series Representations of the Dirac Delta

… ►for all functions $\varphi (x)$ that are continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►provided that $\varphi (x)$ is continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). …

13: 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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14: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important. …

15: 1.9 Calculus of a Complex Variable

… ►The series converges absolutely if ${\sum }_{n=0}^{\mathrm{\infty }}\left|z_n\right|$ converges. A series ${\sum }_{n=0}^{\mathrm{\infty }}{z}_n$ converges (diverges) absolutely when ($>1$), or when ($>1$). Absolutely convergent series are also convergent. … ►The double series is absolutely convergent if it is convergent when ${\zeta }_{m,n}$ is replaced by $\left|{\zeta }_{m,n}\right|$. … ►If a double series is absolutely convergent, then it is also convergent and its sum is given by either of the repeated sums …

16: 15.2 Definitions and Analytical Properties

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Converges absolutely when $\mathrm{\Re }\left(c-a-b\right)>0$.

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17: 16.2 Definition and Analytic Properties

… ►On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\mathrm{\Re }{\gamma }_q>0$, convergent except at $z=1$ if , and divergent if $\mathrm{\Re }{\gamma }_q\le -1$, where …

18: 21.2 Definitions

… ►This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $𝐳$ and $𝛀$ spaces; hence $\theta \left(𝐳\right|𝛀)$ is an analytic function of (each element of) $𝐳$ and (each element of) $𝛀$. …

19: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the $z$-plane. …

20: 8.11 Asymptotic Approximations and Expansions

… ►This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$. …