# convergence properties

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##### 2: 3.3 Interpolation
###### §3.3(vi) Other Interpolation Methods
These references also describe convergence properties of the interpolation formulas. …
##### 3: 3.10 Continued Fractions
can be converted into a continued fraction $C$ of type (3.10.1), and with the property that the $n$th convergent $C_{n}=A_{n}/B_{n}$ to $C$ is equal to the $n$th partial sum of the series in (3.10.3), that is, …
##### 4: 2.5 Mellin Transform Methods
when this integral converges. … To ensure that the integral (2.5.3) converges we assume that … 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. … The extended transform $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ has the same properties as $\mathscr{M}\mskip-3.0muf_{1}\mskip 3.0mu\left(z\right)$ in the half-plane $\Re z. … Next from Table 2.5.1 we observe that the integrals for the transform pair $\mathscr{M}\mskip-3.0muf_{j}\mskip 3.0mu\left(1-z\right)$ and $\mathscr{M}\mskip-3.0muh_{k}\mskip 3.0mu\left(z\right)$ 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). …
##### 6: 21.2 Definitions
21.2.1 $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in% {\mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}% }\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}.$
This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $\mathbf{z}$ and $\boldsymbol{{\Omega}}$ spaces; hence $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is an analytic function of (each element of) $\mathbf{z}$ and (each element of) $\boldsymbol{{\Omega}}$. …
##### 7: 1.3 Determinants, Linear Operators, and Spectral Expansions
###### Relationships Between Determinants
converges1.9(vii)). …Hill-type determinants always converge. …
##### 8: 2.1 Definitions and Elementary Properties
###### §2.1 Definitions and Elementary Properties
Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … The asymptotic property may also hold uniformly with respect to parameters. … As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. … Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …
##### 9: 35.2 Laplace Transform
where the integration variable $\mathbf{X}$ ranges over the space ${\boldsymbol{\Omega}}$. … Then (35.2.1) converges absolutely on the region $\Re\left(\mathbf{Z}\right)>\mathbf{X}_{0}$, and $g(\mathbf{Z})$ is a complex analytic function of all elements $z_{j,k}$ of $\mathbf{Z}$. … Assume that $\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{U}+\mathrm{i}\mathbf{V})\right|% \,\mathrm{d}{\mathbf{V}}$ converges, and also that its limit as $\mathbf{U}\to\infty$ is $0$. …
##### 10: 15.2 Definitions and Analytical Properties
###### §15.2 Definitions and Analytical Properties
On the circle of convergence, $|z|=1$, the Gauss series:
• (a)

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

• ###### §15.2(ii) Analytic Properties
Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …