# pole

(0.001 seconds)

## 21—30 of 58 matching pages

##### 21: 22.10 Maclaurin Series
The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime}$-plane in the case of (22.10.7)–(22.10.9); see §22.17. …
##### 22: 25.2 Definition and Expansions
It is a meromorphic function whose only singularity in $\mathbb{C}$ is a simple pole at $s=1$, with residue 1. …
25.2.4 $\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$
##### 23: Bibliography N
• J. J. Nestor (1984) Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.
• ##### 24: 2.8 Differential Equations with a Parameter
###### §2.8(iv) Case III: Simple Pole
More generally, $g(z)$ can have a simple or double pole at $z_{0}$. (In the case of the double pole the order of the approximating Bessel functions is fixed but no longer $1/(\lambda+2)$.) … For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …
##### 25: 13.4 Integral Representations
where the contour of integration separates the poles of $\Gamma\left(a+t\right)$ from those of $\Gamma\left(-t\right)$. … where the contour of integration separates the poles of $\Gamma\left(a+t\right)\Gamma\left(1+a-b+t\right)$ from those of $\Gamma\left(-t\right)$. …where the contour of integration passes all the poles of $\Gamma\left(b-1+t\right)\Gamma\left(t\right)$ on the right-hand side. …
##### 26: 13.16 Integral Representations
where the contour of integration separates the poles of $\Gamma\left(t-\kappa\right)$ from those of $\Gamma\left(\frac{1}{2}+\mu-t\right)$. … where the contour of integration separates the poles of $\Gamma\left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)$ from those of $\Gamma\left(-\kappa-t\right)$. …where the contour of integration passes all the poles of $\Gamma\left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)$ on the right-hand side.
##### 27: 32.2 Differential Equations
An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. …
##### 28: 19.14 Reduction of General Elliptic Integrals
It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …
##### 29: 22.3 Graphics Figure 22.3.26: Density plot of | sn ⁡ ( 5 , k ) | as a function of complex k 2 , - 10 ≤ ℜ ⁡ ( k 2 ) ≤ 20 , - 10 ≤ ℑ ⁡ ( k 2 ) ≤ 10 . …White spots correspond to poles. Magnify Figure 22.3.27: Density plot of | sn ⁡ ( 10 , k ) | as a function of complex k 2 , - 10 ≤ ℜ ⁡ ( k 2 ) ≤ 20 , - 10 ≤ ℑ ⁡ ( k 2 ) ≤ 10 . …White spots correspond to poles. Magnify Figure 22.3.28: Density plot of | sn ⁡ ( 20 , k ) | as a function of complex k 2 , - 10 ≤ ℜ ⁡ ( k 2 ) ≤ 20 , - 10 ≤ ℑ ⁡ ( k 2 ) ≤ 10 . …White spots correspond to poles. Magnify Figure 22.3.29: Density plot of | sn ⁡ ( 30 , k ) | as a function of complex k 2 , - 10 ≤ ℜ ⁡ ( k 2 ) ≤ 20 , - 10 ≤ ℑ ⁡ ( k 2 ) ≤ 10 . …White spots correspond to poles. Magnify
##### 30: 25.15 Dirichlet $L$-functions
For the principal character $\chi_{1}\pmod{k}$, $L\left(s,\chi_{1}\right)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\phi\left(k\right)/k$, where $\phi\left(k\right)$ is Euler’s totient function (§27.2). …