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expansions in partial fractions


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1: 5.7 Series Expansions
§5.7(ii) Other Series
2: 10.23 Sums
Partial Fractions
For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). …
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: 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.
  • M. Razaz and J. L. Schonfelder (1980) High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.
  • M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
  • D. H. Rouvray (1995) Combinatorics in Chemistry. In Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel, and L. Lovász (Eds.), pp. 1955–1981.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
  • 5: 1.10 Functions of a Complex Variable
    Note that (1.10.4) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6). … If f ( z ) is continuous on D ¯ and analytic in D , then | f ( z ) | attains its maximum on D . … Moreover, if D is bounded and u ( z ) is continuous on D ¯ and harmonic in D , then u ( z ) is maximum at some point on D . …
    §1.10(x) Infinite Partial Fractions
    Mittag-Leffler’s Expansion
    6: Bibliography K
  • E. G. Kalnins and W. Miller (1991a) Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.
  • D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.
  • A. V. Kashevarov (2004) The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.
  • T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
  • I. M. Krichever and S. P. Novikov (1989) Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).
  • 7: 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 ( | z | 1 , z ± i ), whereas the continued fraction (4.25.4) converges for all z except on the branch cuts from i to i and i to i .
    Stieltjes Fractions
    Jacobi Fractions
    8: 33.23 Methods of Computation
    Cancellation errors increase with increases in ρ and | r | , and may be estimated by comparing the final sum of the series with the largest partial sum. … Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). …
    §33.23(v) Continued Fractions
    Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions3.11(iv)) for the analytic continuations of Coulomb functions. …
    9: 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)) … 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. …
    10: Bibliography T
  • N. M. Temme (1979b) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.
  • N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.
  • I. J. Thompson and A. R. Barnett (1985) COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.
  • I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.
  • E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.