# meromorphic

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## 11—20 of 22 matching pages

##### 11: 23.15 Definitions
A modular function $f(\tau)$ is a function of $\tau$ that is meromorphic in the half-plane $\Im\tau>0$, and has the property that for all $\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})$, or for all $\mathcal{A}$ belonging to a subgroup of SL$(2,\mathbb{Z})$, …
##### 12: 25.15 Dirichlet $L$-functions
The notation $L\left(s,\chi\right)$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series …
##### 13: 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 … Furthermore, each $G_{jk}(z)$ has an analytic or meromorphic extension to a half-plane containing $D_{jk}$. …
##### 14: 8.2 Definitions and Basic Properties
When $z\neq 0$, $\Gamma\left(a,z\right)$ is an entire function of $a$, and $\gamma\left(a,z\right)$ is meromorphic with simple poles at $a=-n$, $n=0,1,2,\dots$, with residue $(-1)^{n}/n!$. …
##### 15: 23.3 Differential Equations
As functions of $g_{2}$ and $g_{3}$, $\wp\left(z;g_{2},g_{3}\right)$ and $\zeta\left(z;g_{2},g_{3}\right)$ are meromorphic and $\sigma\left(z;g_{2},g_{3}\right)$ is entire. …
##### 16: 25.16 Mathematical Applications
$H\left(s\right)$ is analytic for $\Re s>1$, and can be extended meromorphically into the half-plane $\Re s>-2k$ for every positive integer $k$ by use of the relations …
##### 17: 29.12 Definitions
In consequence they are doubly-periodic meromorphic functions of $z$. …
##### 18: 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. …
##### 19: 13.2 Definitions and Basic Properties
$M\left(a,b,z\right)$ is entire in $z$ and $a$, and is a meromorphic function of $b$. …
##### 20: 25.11 Hurwitz Zeta Function
$\zeta\left(s,a\right)$ has a meromorphic continuation in the $s$-plane, its only singularity in $\mathbb{C}$ being a simple pole at $s=1$ with residue $1$. …