expansions in partial fractions
1—10 of 14 matching pages
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… ►
4: Bibliography R
A code to calculate (high order) Bessel functions based on the continued fractions method.
Comput. Phys. Comm. 76 (3), pp. 381–388.
High precision Chebyshev expansions for Airy functions and their derivatives.
University of Birmingham Computer Centre.
Partial fractions expansions and identities for products of Bessel functions.
J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
Combinatorics in Chemistry.
In Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel, and L. Lovász (Eds.),
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,
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 is continuous on and analytic in , then attains its maximum on . … ►Moreover, if is bounded and is continuous on and harmonic in , then is maximum at some point on . … ►
§1.10(x) Infinite Partial Fractions… ►
6: Bibliography K
Hypergeometric expansions of Heun polynomials.
SIAM J. Math. Anal. 22 (5), pp. 1450–1459.
Series expansions for the third incomplete elliptic integral via partial fraction decompositions.
J. Comput. Appl. Math. 207 (2), pp. 331–337.
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.
Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators.
SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
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 of type (3.10.1), and with the property that the th convergent to is equal to the 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 (4.24.3), we obtain a continued fraction with the same region of convergence (, ), whereas the continued fraction (4.25.4) converges for all except on the branch cuts from to and to . ►
Stieltjes Fractions… ►
8: 33.23 Methods of Computation
… ►Cancellation errors increase with increases in and , 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 expansions (§3.11(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 fractions (§1.2(iii)) … ►By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of for large , or small , can be obtained complete with an integral representation of the error term. …
10: 2.4 Contour Integrals
… ►In consequence, the asymptotic expansion obtained from (2.4.14) is no longer null. … ►The final expansion then has the form … ►Suppose that on the integration path there are two simple zeros of that coincide for a certain value of . … ►with and chosen so that the zeros of correspond to the zeros , say, of the quadratic . … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …