# expansions in doubly-infinite partial fractions

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##### 1: 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

►With $t\in \u2102$ and …The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important. … ►
22.12.13
$$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty}}{lim}\sum _{n=-N}^{N}{(-1)}^{n}\frac{\pi}{\mathrm{tan}\left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty}}{lim}\sum _{n=-N}^{N}{(-1)}^{n}\left(\underset{M\to \mathrm{\infty}}{lim}\sum _{m=-M}^{M}\frac{1}{t-m-n\tau}\right).$$

##### 2: 5.19 Mathematical Applications

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►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions.
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►By decomposition into partial fractions (§1.2(iii))
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►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. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …##### 3: 31.11 Expansions in Series of Hypergeometric Functions

###### §31.11 Expansions in Series of Hypergeometric Functions

… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►###### §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.##### 4: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

###### §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. … ►where the coefficients are as in §28.14.##### 5: 1.9 Calculus of a Complex Variable

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►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 $S$.
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►Then the expansions (1.9.54), (1.9.57), and (1.9.60) hold for all sufficiently small $|z|$.
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##### 6: 28.11 Expansions in Series of Mathieu Functions

###### §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). …See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of $q$ see Meixner et al. (1980, p. 33). … ►
28.11.7
$$\mathrm{sin}(2m+2)z=\sum _{n=0}^{\mathrm{\infty}}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

##### 7: 36.9 Integral Identities

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►For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980).
This reference also provides a physical interpretation in terms of Lagrangian manifolds and Wigner functions in phase space.
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##### 8: 28.29 Definitions and Basic Properties

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$Q(z)$ 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.
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►The

*basic solutions*${w}_{\text{I}}(z,\lambda )$, ${w}_{\text{II}}(z,\lambda )$ are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). … ►###### §28.29(iii) Discriminant and Eigenvalues in the Real Case

… ►Assume that the second derivative of $Q(x)$ in (28.29.1) exists and is continuous. … ►For further results, especially when $Q(z)$ is analytic in a strip, see Weinstein and Keller (1987).##### 9: 16.22 Asymptotic Expansions

###### §16.22 Asymptotic Expansions

►Asymptotic expansions of ${G}_{p,q}^{m,n}(z;\mathbf{a};\mathbf{b})$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).##### 10: Howard S. Cohl

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► 1968 in Paterson, New Jersey) is a Mathematician in the Applied and Computational Mathematics Division at the National Institute of Standards and Technology in Gaithersburg, Maryland.
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► in astronomy and astrophysics from Indiana University, Bloomington, Indiana, a M.
… in physics from Louisiana State University, Baton Rouge, Louisiana, and a Ph.
…in mathematics from the University of Auckland in New Zealand.
►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series.
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