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

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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:

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See also:

Annotations for §22.12 and Ch.22

2: 5.19 Mathematical Applications

… ►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. … ►By decomposition into partial fractions (§1.2(iii)) … ►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

… ► … ► … ►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 $S$. … ►Then the expansions (1.9.54), (1.9.57), and (1.9.60) hold for all sufficiently small $\left|z\right|$. …

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 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

7: 36.9 Integral Identities

… ►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. …

8: 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. … ►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 $G_{p,q}^{m,n}(z;𝐚;𝐛)$ 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: 3.10 Continued Fractions

… ►can be converted into a continued fraction $C$ of type (3.10.1), and with the property that the $n$th convergent $C_n=A_n/B_n$ to $C$ is equal to the $n$th partial sum of the series in (3.10.3), that is, … ►However, other continued fractions with the same limit may converge in a much larger domain of the complex plane than the fraction given by (3.10.4) and (3.10.5). For example, by converting the Maclaurin expansion of $\arctan z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in ℂ$ except on the branch cuts from $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$ and $-\mathrm{i}$ to $-\mathrm{i}\mathrm{\infty }$. … ►if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$ agrees with (3.10.7) up to and including the term in $z^{n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. …