convergence properties

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1: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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Convergence Properties

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2: 3.3 Interpolation

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§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 $ℳf\left(z\right)$ has the same properties as $ℳf_1\left(z\right)$ in the half-plane . … ►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). …

5: 16.2 Definition and Analytic Properties

§16.2 Definition and Analytic Properties

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§16.2(ii) Case $p\le q$

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§16.2(iii) Case $p=q+1$

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§16.2(iv) Case $p>q+1$

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§16.2(v) Behavior with Respect to Parameters

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6: 21.2 Definitions

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21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

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Defines:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(i), §21.2 and Ch.21

►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) $𝛀$. …

7: 1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(i) Determinants: Elementary Properties

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Relationships Between Determinants

… ► … ►converges (§1.9(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 $𝐗$ ranges over the space $𝛀$. … ►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 $𝐙$. … ►Assume that ${\int }_{𝓢}\left|g(𝐔+\mathrm{i}𝐕)\right|d𝐕$ converges, and also that its limit as $𝐔\to \mathrm{\infty }$ 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: ►

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

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§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. …