doubly-infinite

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1: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. …

2: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

3: 5.19 Mathematical Applications

… ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …

4: 28.11 Expansions in Series of Mathieu Functions

… ►Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-infinite strip $S$ that contains the real axis, and $q$ be a normal value (§28.7). …

5: 31.11 Expansions in Series of Hypergeometric Functions

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§31.11(v) Doubly-Infinite Series

►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …

6: 36.9 Integral Identities

… ►For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). …

7: 28.29 Definitions and Basic Properties

… ► $Q(z)$ is either a continuous and real-valued function for $z\in ℝ$ or an analytic function of $z$ in a doubly-infinite open strip that contains the real axis. …

8: 1.9 Calculus of a Complex Variable

… ►A doubly-infinite series ${\sum }_{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n(z)$ converges (uniformly) on $S$ iff each of the series ${\sum }_{n=0}^{\mathrm{\infty }}{f}_n(z)$ and ${\sum }_{n=1}^{\mathrm{\infty }}{f}_{-n}(z)$ converges (uniformly) on $S$. …