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21: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. …

22: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

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§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

… ► $F_{\mathrm{\ell }}(\eta ,\rho )$ is a real and analytic function of $\rho$ on the open interval , and also an analytic function of $\eta$ when . … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . Also, ${\mathrm{e}}^{\mp \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{H}_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are analytic functions of $\eta$ when . …

23: 16.15 Integral Representations and Integrals

… ►These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large $x$, large $y$, or both. …

24: 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 $𝐙$. …

25: 23.15 Definitions

… ►If, as a function of $q$, $f(\tau )$ is analytic at $q=0$, then $f(\tau )$ is called a modular form. … …

26: 25.15 Dirichlet $L$-functions

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§25.15(i) Definitions and Basic Properties

… ►For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). …

27: 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:

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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) $𝛀$. …

28: 28.12 Definitions and Basic Properties

… ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization … ►

§28.12(iii) Functions ${\mathrm{ce}}_{\nu }(z,q)$, ${\mathrm{se}}_{\nu }(z,q)$, when $\nu \notin \mathbb{Z}$

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29: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

… ► $f(ϵ,\mathrm{\ell };r)$ is real and an analytic function of $r$ in the interval , and it is also an analytic function of $ϵ$ when . … ►

§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

… ► $h(ϵ,\mathrm{\ell };r)$ is real and an analytic function of each of $r$ and $ϵ$ in the intervals and , except when $r=0$ or $ϵ=0$. …

30: 2.10 Sums and Sequences

… ►Let $f(z)$ be analytic on the annulus , with Laurent expansion ►

2.10.25 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{z}^n,$$ .

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Symbols:

$f(x)$: analytic function, $f_n$: coefficients and $r$: radious of annulus

Permalink:

http://dlmf.nist.gov/2.10.E25

Encodings:

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See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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2.10.26 $$f_n=\frac{1}{2\pi \mathrm{i}}{\int }_{𝒞}\frac{f(z)}{z^{n+1}}dz,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $f(x)$: analytic function, $f_n$: coefficients and $𝒞$: simple closed contour

Permalink:

http://dlmf.nist.gov/2.10.E26

Encodings:

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See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

2.10.30 $$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$ $z\to z_j$,

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E30

Encodings:

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See also:

Annotations for §2.10(iv), §2.10 and Ch.2

where ${\sigma }_j$ is a positive constant.

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2.10.32 $$f^{(m)}(z)-g^{(m)}(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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