absolutely convergent

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21: 11.2 Definitions

… ►The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of $z$. …

22: 28.4 Fourier Series

… ►The Fourier series of the periodic Mathieu functions converge absolutely and uniformly on all compact sets in the $z$-plane. …

23: 2.4 Contour Integrals

… ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►

$z$ ranges along a ray or over an annular sector ${\theta }_1\le \theta \le {\theta }_2$, $|z|\ge Z$, where $\theta =\mathrm{ph}z$, , and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

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24: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►If $p=q+1$, then (35.8.1) converges absolutely for and diverges for $\Vert 𝐓\Vert >1$. …

25: 4.13 Lambert $W$-Function

… ►For large enough $\left|z\right|$ the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. …

26: 1.10 Functions of a Complex Variable

… ►The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. … ►If $\left|f(z,t)\right|\le M(t)$ for $z\in S$ and ${\int }_a^{b}M(t)dt$ converges, then the integral (1.10.18) converges uniformly and absolutely in $S$. … ►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. …

27: 2.5 Mellin Transform Methods

… ►With these definitions and the conditions (2.5.17)–(2.5.20) the Mellin transforms converge absolutely and define analytic functions in the half-planes shown in Table 2.5.1. … ►Next from Table 2.5.1 we observe that the integrals for the transform pair $ℳf_j\left(1-z\right)$ and $ℳh_k\left(z\right)$ are absolutely convergent in the domain $D_{jk}$ specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20). …

28: 1.5 Calculus of Two or More Variables

… ►whenever both repeated integrals exist and at least one is absolutely convergent. …

29: 2.3 Integrals of a Real Variable

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The integral (2.3.13) converges absolutely for all sufficiently large $x$.

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30: 11.10 Anger–Weber Functions

… ►These expansions converge absolutely for all finite values of $z$. …