# around infinity

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## 7 matching pages

##### 2: 25.5 Integral Representations
where the integration contour is a loop around the negative real axis; it starts at $-\infty$, encircles the origin once in the positive direction without enclosing any of the points $z=\pm 2\pi\mathrm{i}$, $\pm 4\pi\mathrm{i}$, …, and returns to $-\infty$. …
##### 3: 2.10 Sums and Sequences
(5.11.7) shows that the integrals around the large quarter circles vanish as $n\to\infty$. …
##### 4: 5.11 Asymptotic Expansions
5.11.3 $\Gamma\left(z\right)={\mathrm{e}}^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}% \Gamma^{*}\left(z\right)\sim{\mathrm{e}}^{-z}z^{z}\left(\frac{2\pi}{z}\right)^% {1/2}\sum_{k=0}^{\infty}\frac{g_{k}}{z^{k}},$
##### 5: 25.11 Hurwitz Zeta Function
where the integration contour (see Figure 5.9.1) is a loop around the negative real axis as described for (25.5.20). …
25.11.39 $\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,\tfrac{3}{4}\right)=8G,$
As $\beta\to\pm\infty$ with $s$ fixed, $\Re s>1$, … As $a\to\infty$ in the sector $|\operatorname{ph}a|\leq\pi-\delta(<\pi)$, with $s(\neq 1)$ and $\delta$ fixed, we have the asymptotic expansion … Similarly, as $a\to\infty$ in the sector $|\operatorname{ph}a|\leq\pi-\delta(<\pi)$. …
##### 6: 1.10 Functions of a Complex Variable
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. … where $N$ and $P$ are respectively the numbers of zeros and poles, counting multiplicity, of $f$ within $C$, and $\Delta_{C}(\operatorname{ph}f(z))$ is the change in any continuous branch of $\operatorname{ph}\left(f(z)\right)$ as $z$ passes once around $C$ in the positive sense. … Branches of $F(z)$ can be defined, for example, in the cut plane $D$ obtained from $\mathbb{C}$ by removing the real axis from $1$ to $\infty$ and from $-1$ to $-\infty$; see Figure 1.10.1. … The product $\prod^{\infty}_{n=1}(1+a_{n})$, with $a_{n}\not=-1$ for all $n$, converges iff $\sum^{\infty}_{n=1}\ln\left(1+a_{n}\right)$ converges; and it converges absolutely iff $\sum^{\infty}_{n=1}\left|a_{n}\right|$ converges. …
##### 7: 18.39 Applications in the Physical Sciences
allows anharmonic, or amplitude dependent, frequencies of oscillation about $x_{e}$, and also escape of the particle to $x=+\infty$ with dissociation energy $D$. … where the orthogonality measure is now $\,\mathrm{d}r$, $r\in[0,\infty).$Orthogonality, with measure $\,\mathrm{d}r$ for $r\in[0,\infty)$, for fixed $l$normalized with measure $r^{2}\,\mathrm{d}r$, $r\in[0,\infty)$. … For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983). …