meromorphic function

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1: 21.8 Abelian Functions

… ►An Abelian function is a $2g$-fold periodic, meromorphic function of $g$ complex variables. …

2: 13.5 Continued Fractions

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

3: 13.17 Continued Fractions

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

4: 22.17 Moduli Outside the Interval [0,1]

… ►When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^2$. …

5: 5.2 Definitions

… ►It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^n/n!$ at $z=-n$. …

6: 1.10 Functions of a Complex Variable

… ►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. … …

7: 2.5 Mellin Transform Methods

… ►If $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ can be continued analytically to meromorphic functions in a left half-plane, and if the contour $\mathrm{\Re }z=c$ can be translated to $\mathrm{\Re }z=d$ with , then …Similarly, if $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for $I(x)$ for large values of $x$. … ►Furthermore, $ℳf_1\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane, whose singularities are simple poles at $-{\alpha }_s$, $s=0,1,2,\mathrm{\dots }$, with principal part … ►Similarly, if $\kappa =0$ in (2.5.18), then $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane with simple poles at ${\beta }_s$, $s=0,1,2,\mathrm{\dots }$, with principal part … ►Similarly, since $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function (when $\kappa =0$) or to an entire function (when $\kappa \ne 0$), we can choose $\rho$ so that $ℳh_2\left(z\right)$ has no poles in . …

8: 25.2 Definition and Expansions

… ►It is a meromorphic function whose only singularity in $ℂ$ is a simple pole at $s=1$, with residue 1. … ►

25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

…

9: 23.2 Definitions and Periodic Properties

… ► $\mathrm{\wp }\left(z\right)$ and $\zeta \left(z\right)$ are meromorphic functions with poles at the lattice points. … ►Hence $\mathrm{\wp }\left(z\right)$ is an elliptic function, that is, $\mathrm{\wp }\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\mathrm{\wp }\left(z\right)$ is meromorphic and has two periods whose ratio is not real. …

10: 8.21 Generalized Sine and Cosine Integrals

… ►Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. …