# pole

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## 11—20 of 59 matching pages

##### 11: 4.14 Definitions and Periodicity
The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …
##### 12: 7.20 Mathematical Applications
For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). …
##### 13: 8.15 Sums
8.15.2 $a\sum_{k=1}^{\infty}\left(\frac{{\mathrm{e}}^{2\pi\mathrm{i}k(z+h)}}{\left(2% \pi\mathrm{i}k\right)^{a+1}}\Gamma\left(a,2\pi\mathrm{i}kz\right)+\frac{{% \mathrm{e}}^{-2\pi\mathrm{i}k(z+h)}}{\left(-2\pi\mathrm{i}k\right)^{a+1}}% \Gamma\left(a,-2\pi\mathrm{i}kz\right)\right)=\zeta\left(-a,z+h\right)+\frac{z% ^{a+1}}{a+1}+\left(h-\tfrac{1}{2}\right)z^{a},$ $h\in[0,1]$.
##### 14: 23.2 Definitions and Periodic Properties
$\wp\left(z\right)$ and $\zeta\left(z\right)$ are meromorphic functions with poles at the lattice points. …The poles of $\wp\left(z\right)$ are double with residue $0$; the poles of $\zeta\left(z\right)$ are simple with residue $1$. … …
##### 15: Bibliography D
• B. Deconinck and H. Segur (2000) Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.
• T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
• T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.
• T. M. Dunster (2004) Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.
• ##### 16: 33.22 Particle Scattering and Atomic and Molecular Spectra
###### §33.22(vii) Complex Variables and Parameters
• Searches for resonances as poles of the $S$-matrix in the complex half-plane $\Im\sf{k}<0$. See for example Csótó and Hale (1997).

• Regge poles at complex values of $\ell$. See for example Takemasa et al. (1979).

• ##### 17: 12.5 Integral Representations
where the contour separates the poles of $\Gamma\left(t\right)$ from those of $\Gamma\left(\tfrac{1}{2}+a-2t\right)$. … where the contour separates the poles of $\Gamma\left(t\right)$ from those of $\Gamma\left(\tfrac{1}{2}-a-2t\right)$. …
##### 18: 16.11 Asymptotic Expansions
It may be observed that $H_{p,q}(z)$ represents the sum of the residues of the poles of the integrand in (16.5.1) at $s=-a_{j},-a_{j}-1,\dots$, $j=1,\dots,p$, provided that these poles are all simple, that is, no two of the $a_{j}$ differ by an integer. (If this condition is violated, then the definition of $H_{p,q}(z)$ has to be modified so that the residues are those associated with the multiple poles. …
##### 19: 14.21 Definitions and Basic Properties
$P^{\pm\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\infty$, which are branch points (or poles) of the functions, in general. …
##### 20: 15.3 Graphics Figure 15.3.6: F ⁡ ( - 3 , 3 5 ; u + i ⁢ v ; 1 2 ) , - 6 ≤ u ≤ 2 , - 2 ≤ v ≤ 2 . (With c = u + i ⁢ v the only poles occur at c = 0 , - 1 , - 2 ; compare §15.2(ii).) Magnify 3D Help