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infinite partial fractions

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1: 4.36 Infinite Products and Partial Fractions
§4.36 Infinite Products and Partial Fractions
2: 4.22 Infinite Products and Partial Fractions
§4.22 Infinite Products and Partial Fractions
3: 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 2 K cs ( 2 K t , k ) = lim N n = - N N ( - 1 ) n π tan ( π ( t - n τ ) ) = lim N n = - N N ( - 1 ) n ( lim M m = - M M 1 t - m - n τ ) .
4: 1.10 Functions of a Complex Variable
§1.10(x) Infinite Partial Fractions
Mittag-Leffler’s Expansion
5: 1.2 Elementary Algebra
§1.2(iii) Partial Fractions
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 Ai ( ± x ) and Bi ( ± 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 ω is the angle of slope of 𝒫 at a , that is, lim ( ph ( t - a ) ) as t a along 𝒫 . … However, if p ( t 0 ) = 0 , then μ 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 𝒫 there are two simple zeros of p ( α , t ) / t that coincide for a certain value α ^ of α . … with a and b chosen so that the zeros of p ( α , t ) / t correspond to the zeros w 1 ( α ) , w 2 ( α ) , say, of the quadratic w 2 + 2 a w + b . …
    2.4.20 f ( α , w ) = q ( α , t ) d t d w = q ( α , t ) w 2 + 2 a w + b p ( α , t ) / t .