# meromorphic function

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## 1—10 of 22 matching pages

##### 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 $\mathbb{C}$. 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 $|\operatorname{ph}{z}|<\pi$. …
##### 3: 13.17 Continued Fractions
This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in\mathbb{C}$. 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 $|\operatorname{ph}{z}|<\pi$. …
##### 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 $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ can be continued analytically to meromorphic functions in a left half-plane, and if the contour $\Re z=c$ can be translated to $\Re z=d$ with $d, then …Similarly, if $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ 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, $\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane, whose singularities are simple poles at $-\alpha_{s}$, $s=0,1,2,\dots$, with principal part … Similarly, if $\kappa=0$ in (2.5.18), then $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane with simple poles at $\beta_{s}$, $s=0,1,2,\dots$, with principal part … Similarly, since $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ can be continued analytically to a meromorphic function (when $\kappa=0$) or to an entire function (when $\kappa\neq 0$), we can choose $\rho$ so that $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ has no poles in $1<\Re z\leq\rho<2$. …
##### 8: 25.2 Definition and Expansions
It is a meromorphic function whose only singularity in $\mathbb{C}$ is a simple pole at $s=1$, with residue 1. …
25.2.4 $\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$
##### 9: 23.2 Definitions and Periodic Properties
$\wp\left(z\right)$ and $\zeta\left(z\right)$ are meromorphic functions with poles at the lattice points. … Hence $\wp\left(z\right)$ is an elliptic function, that is, $\wp\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\wp\left(z\right)$ is meromorphic and has two periods whose ratio is not real. …
##### 10: 8.21 Generalized Sine and Cosine Integrals
Furthermore, $\mathrm{si}\left(a,z\right)$ and $\mathrm{ci}\left(a,z\right)$ are entire functions of $a$, and $\mathrm{Si}\left(a,z\right)$ and $\mathrm{Ci}\left(a,z\right)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\dots$ and $a=0,-2,-4,\dots$, respectively. …