# infinite partial fractions

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##### 3: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
###### §22.12 Expansions in Other Trigonometric Series and Doubly-InfinitePartialFractions: Eisenstein Series
22.12.13 $2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}% \frac{\pi}{\tan\left(\pi(t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)% ^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right).$
##### 6: 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 fractions1.2(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. …
##### 7: Bibliography R
• Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.
• J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
• M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
• M. Rothman (1954b) The problem of an infinite plate under an inclined loading, with tables of the integrals of $\rm{Ai}(\pm x)$ and $\rm{Bi}(\pm x)$ . Quart. J. Mech. Appl. Math. 7 (1), pp. 1–7.
• ##### 8: 2.4 Contour Integrals
in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $\mathscr{P}$ at $a$, that is, $\lim(\operatorname{ph}\left(t-a\right))$ as $t\to a$ along $\mathscr{P}$. … However, if $p^{\prime}(t_{0})=0$, then $\mu\geq 2$ and different branches of some of the fractional powers of $p_{0}$ are used for the coefficients $b_{s}$; again see §2.3(iii). … Suppose that on the integration path $\mathscr{P}$ there are two simple zeros of $\ifrac{\partial p(\alpha,t)}{\partial t}$ that coincide for a certain value $\widehat{\alpha}$ of $\alpha$. … with $a$ and $b$ chosen so that the zeros of $\ifrac{\partial p(\alpha,t)}{\partial t}$ correspond to the zeros $w_{1}(\alpha),w_{2}(\alpha)$, say, of the quadratic $w^{2}+2aw+b$. …
2.4.20 $f(\alpha,w)=q(\alpha,t)\frac{\mathrm{d}t}{\mathrm{d}w}=q(\alpha,t)\frac{w^{2}+% 2aw+b}{\ifrac{\partial p(\alpha,t)}{\partial t}}.$