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

… ►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. … ►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. …

2: 10.72 Mathematical Applications

… ►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 $z_0$. … ►

§10.72(ii) Differential Equations with Poles

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§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

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§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

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5: 13.27 Mathematical Applications

… ►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). …

6: 8.6 Integral Representations

… ►where the integration path passes above or below the pole at $t=1$, according as upper or lower signs are taken. … ►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 $s=0,1,2,\mathrm{\dots }$. …

7: 1.10 Functions of a Complex Variable

… ►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

… ►where the integration path $L$ separates the poles of the factors $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ from those of the factors $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$. … ►

$L$ 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 $p=q\ge 1$.

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$L$ 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 $p=q\ge 1$.

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9: 5.2 Definitions

… ►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 $z=-n$. …

10: 32.11 Asymptotic Approximations for Real Variables

… ►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. … ►If $|k|>1$, then $w_k(x)$ has a pole at a finite point $x=c_0$, dependent on $k$, and … ►then $w_h(x)$ has no poles on the real axis. … ►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 $h$. …