# meromorphic function

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## 1—10 of 22 matching pages

##### 1: 21.8 Abelian Functions

##### 2: 13.5 Continued Fractions

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►This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $\u2102$.
For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).
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►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $$.
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##### 3: 13.17 Continued Fractions

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►This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in \u2102$.
For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).
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►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $$.
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##### 4: 22.17 Moduli Outside the Interval [0,1]

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►When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of ${k}^{2}$.
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##### 5: 5.2 Definitions

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►It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^{n}/n!$ at $z=-n$.
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##### 6: 1.10 Functions of a Complex Variable

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

*meromorphic function*. … …##### 7: 2.5 Mellin Transform Methods

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►If $\mathcal{M}f\left(1-z\right)$ and $\mathcal{M}h\left(z\right)$ 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 $$, then
…Similarly, if $\mathcal{M}f\left(1-z\right)$ and $\mathcal{M}h\left(z\right)$ can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for $I(x)$ for large values of $x$.
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►Furthermore, $\mathcal{M}{f}_{1}\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane, whose singularities are simple poles at $-{\alpha}_{s}$, $s=0,1,2,\mathrm{\dots}$, with principal part
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►Similarly, if $\kappa =0$ in (2.5.18), then $\mathcal{M}{h}_{2}\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane with simple poles at ${\beta}_{s}$, $s=0,1,2,\mathrm{\dots}$, with principal part
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►Similarly, since $\mathcal{M}{h}_{2}\left(z\right)$ can be continued analytically to a meromorphic function (when $\kappa =0$) or to an entire function (when $\kappa \ne 0$), we can choose $\rho $ so that $\mathcal{M}{h}_{2}\left(z\right)$ has no poles in $$.
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##### 8: 25.2 Definition and Expansions

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►It is a meromorphic function whose only singularity in $\u2102$ is a simple pole at $s=1$, with residue 1.
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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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##### 9: 23.2 Definitions and Periodic Properties

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$\mathrm{\wp}\left(z\right)$ and $\zeta \left(z\right)$ are meromorphic functions with poles at the lattice points.
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►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

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►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.
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