# doubly-infinite

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## 8 matching pages

##### 1: 28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ 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
22.12.13 $2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}% \frac{\pi}{\tan\left(\pi(t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)% ^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right).$
##### 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
###### §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 $z\in\mathbb{R}$ 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 $\sum^{\infty}_{n=-\infty}f_{n}(z)$ converges (uniformly) on $S$ iff each of the series $\sum^{\infty}_{n=0}f_{n}(z)$ and $\sum^{\infty}_{n=1}f_{-n}(z)$ converges (uniformly) on $S$. …