# poles

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## 1—10 of 60 matching pages

##### 1: Sidebar 5.SB1: Gamma & Digamma Phase Plots

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►In the upper half of the image, the poles of $\mathrm{\Gamma}\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi $ around each pole, showing a full revolution of the color wheel.
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►In the lower half of the image, the poles of $\psi \left(z\right)$ (corresponding to the poles of $\mathrm{\Gamma}\left(z\right)$) and the zeros between them are clear.
Phase changes around the zeros are of opposite sign to those around the poles.
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##### 2: 10.72 Mathematical Applications

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►The number $m$ can also be replaced by any real constant $\lambda $
$(>-2)$ in the sense that ${(z-{z}_{0})}^{-\lambda}$
$f(z)$ is analytic and nonvanishing at ${z}_{0}$; moreover, $g(z)$ is permitted to have a single or double pole at ${z}_{0}$.
The order of the approximating Bessel functions, or modified Bessel functions, is $1/(\lambda +2)$, except in the case when $g(z)$ has a double pole at ${z}_{0}$.
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###### §10.72(ii) Differential Equations with Poles

… ►###### §10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point

►In (10.72.1) assume $f(z)=f(z,\alpha )$ and $g(z)=g(z,\alpha )$ depend continuously on a real parameter $\alpha $, $f(z,\alpha )$ has a simple zero $z={z}_{0}(\alpha )$ and a double pole $z=0$, except for a critical value $\alpha =a$, where ${z}_{0}(a)=0$. …##### 3: 22.4 Periods, Poles, and Zeros

###### §22.4 Periods, Poles, and Zeros

►###### §22.4(i) Distribution

►For each Jacobian function, Table 22.4.1 gives its periods in the $z$-plane in the left column, and the position of one of its poles in the second row. … ►The other poles and zeros are at the congruent points. … ►Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. …##### 4: 4.28 Definitions and Periodicity

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###### Periodicity and Zeros

…##### 5: 13.27 Mathematical Applications

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►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).
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##### 6: 8.6 Integral Representations

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►where the integration path passes above or below the pole at $t=1$, according as upper or lower signs are taken.
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►In (8.6.10)–(8.6.12), $c$ is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at $s=a$, in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots}$.
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##### 7: 1.10 Functions of a Complex Variable

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►Lastly, if ${a}_{n}\ne 0$ for infinitely many negative $n$, then ${z}_{0}$ is an

*isolated essential singularity*. … ►A function whose only singularities, other than the point at infinity, are poles is called a*meromorphic function*. If the poles are infinite in number, then the point at infinity is called an*essential singularity*: it is the limit point of the poles. … ►If the singularities within $C$ are poles and $f(z)$ is analytic and nonvanishing on $C$, then … ►each location again being counted with multiplicity equal to that of the corresponding zero or pole. …##### 8: 16.17 Definition

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►where the integration path $L$ separates the poles of the factors $\mathrm{\Gamma}\left({b}_{\mathrm{\ell}}-s\right)$ from those of the factors $\mathrm{\Gamma}\left(1-{a}_{\mathrm{\ell}}+s\right)$.
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(ii)
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(iii)
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$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\mathrm{\Gamma}\left({b}_{\mathrm{\ell}}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ($\ne 0$) if $$, and for $$ if $p=q\ge 1$.

$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\mathrm{\Gamma}\left(1-{a}_{\mathrm{\ell}}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all $z$ if $p>q$, and for $|z|>1$ if $p=q\ge 1$.

##### 9: 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 $z=-n$.
…$\psi \left(z\right)$ is meromorphic with simple poles of residue $-1$ at $z=-n$.
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##### 10: 32.11 Asymptotic Approximations for Real Variables

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►Next, for given initial conditions $w(0)=0$ and ${w}^{\prime}(0)=k$, with $k$ real, $w(x)$ has at least one pole on the real axis.
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►If $|k|>1$, then ${w}_{k}(x)$ has a pole at a finite point $x={c}_{0}$, dependent on $k$, and
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►then ${w}_{h}(x)$ has no poles on the real axis.
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►and ${w}_{{h}^{\ast}}(x)$ has no poles on the real axis.
►Lastly if $h>{h}^{\ast}$, then ${w}_{h}(x)$ has a simple pole on the real axis, whose location is dependent on $h$.
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