absolutely convergent

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1: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►The series (28.19.2) converges absolutely and uniformly on compact subsets within $S$. …

2: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series …

3: 14.13 Trigonometric Expansions

… ►These Fourier series converge absolutely when . If then they converge, but, if $\theta \ne \frac{1}{2}\pi$, they do not converge absolutely. …

4: 27.4 Euler Products and Dirichlet Series

… ►if the series on the left is absolutely convergent. In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the Euler product of the series. …

5: 27.7 Lambert Series as Generating Functions

… ►If , then the quotient $x^n/(1-x^n)$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: …

6: 35.2 Laplace Transform

… ►Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. …

7: 27.5 Inversion Formulas

… ►

27.5.7 $$G(x)=\sum _{m=1}^{\mathrm{\infty }}\frac{F(mx)}{m^s}⟺F(x)=\sum _{m=1}^{\mathrm{\infty }}\mu \left(m\right)\frac{G(mx)}{m^s},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $m$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

Referenced by:

§27.17, §27.5

Permalink:

http://dlmf.nist.gov/27.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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8: 28.11 Expansions in Series of Mathieu Functions

… ►The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip $S$. …

9: 28.14 Fourier Series

… ►converge absolutely and uniformly on all compact sets in the $z$-plane. …

10: 1.4 Calculus of One Variable

… ►If the limits exist with $f(x)$ replaced by $\left|f(x)\right|$, then the integrals are absolutely convergent. … ► …