meromorphic

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11: 23.15 Definitions

… ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, …

12: 25.15 Dirichlet $L$-functions

… ►The notation $L(s,\chi )$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series …

13: 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 … ►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 $z\ne 0$, $\mathrm{\Gamma }(a,z)$ is an entire function of $a$, and $\gamma (a,z)$ is meromorphic with simple poles at $a=-n$, $n=0,1,2,\mathrm{\dots }$, with residue ${(-1)}^n/n!$. …

15: 23.3 Differential Equations

… ►As functions of $g_2$ and $g_3$, $\mathrm{\wp }(z;g_2,g_3)$ and $\zeta (z;g_2,g_3)$ are meromorphic and $\sigma (z;g_2,g_3)$ is entire. …

16: 25.16 Mathematical Applications

… ► $H\left(s\right)$ is analytic for $\mathrm{\Re }s>1$, and can be extended meromorphically into the half-plane $\mathrm{\Re }s>-2k$ 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, $\mathrm{si}(a,z)$ 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. …

19: 13.2 Definitions and Basic Properties

… ► $M(a,b,z)$ is entire in $z$ and $a$, and is a meromorphic function of $b$. …

20: 25.11 Hurwitz Zeta Function

… ► $\zeta (s,a)$ has a meromorphic continuation in the $s$-plane, its only singularity in $ℂ$ being a simple pole at $s=1$ with residue $1$. …