around infinity

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1: 1.9 Calculus of a Complex Variable

… ►A system of open disks around infinity is given by ►

1.9.35 $$S_r=\{z\mid |z|>1/r\}\cup \{\mathrm{\infty }\},$$ .

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Symbols:

$\cup$: union, $z$: variable, $r$: radius and $S_r$: neighborhood

Permalink:

http://dlmf.nist.gov/1.9.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(iv), §1.9 and Ch.1

…

2: 25.5 Integral Representations

… ►where the integration contour is a loop around the negative real axis; it starts at $-\mathrm{\infty }$, 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 $-\mathrm{\infty }$. …

3: 2.10 Sums and Sequences

… ►(5.11.7) shows that the integrals around the large quarter circles vanish as $n\to \mathrm{\infty }$. …

4: 5.11 Asymptotic Expansions

… ►

5.11.3 $$\mathrm{\Gamma }\left(z\right)\sim {\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{g_k}{z^k},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable and $g_k$: coefficients

A&S Ref:

6.1.37

Referenced by:

§10.19(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §8.11(ii), §8.12

Permalink:

http://dlmf.nist.gov/5.11.E3

Encodings:

pMML, png, TeX

Changes (effective with 1.0.11):

An unnecessary bracket around the sum was removed.

See also:

Annotations for §5.11(i), §5.11 and Ch.5

…

5: 25.11 Hurwitz Zeta Function

… ►where the integration contour is a loop around the negative real axis as described for (25.5.20). … ►

25.11.39 $$\sum _{k=2}^{\mathrm{\infty }}\frac{k}{2^k}\zeta (k+1,\frac{3}{4})=8G,$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer and $G$: Catalan’s constant

Keywords:

Catalan’s constant, infinite series

Source:

Adamchik and Srivastava (1998, (2.30), p. 138)

Permalink:

http://dlmf.nist.gov/25.11.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xi), §25.11 and Ch.25

… ►As $\beta \to \pm \mathrm{\infty }$ with $s$ fixed, $\mathrm{\Re }s>1$, … ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

6: 18.39 Physical Applications

… ►For (18.39.2) to have a nontrivial bounded solution in the interval , the constant $E$ (the total energy of the particle) must satisfy … ►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). …

7: 1.10 Functions of a Complex Variable

… ►The singularities of $f(z)$ at infinity are classified in the same way as the singularities of $f(1/z)$ at $z=0$. … ►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 ${\mathrm{\Delta }}_C(\mathrm{ph}f(z))$ is the change in any continuous branch of $\mathrm{ph}\left(f(z)\right)$ 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 $ℂ$ by removing the real axis from $1$ to $\mathrm{\infty }$ and from $-1$ to $-\mathrm{\infty }$; see Figure 1.10.1. …