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Orthogonal Polynomials Arising in the Numerical Evaluation of Inverse Laplace Transforms
Herbert E. Salzer
Mathematical Tables and Other Aids to Computation
Vol. 9, No. 52 (Oct., 1955), pp. 164177
Published by: American Mathematical Society
DOI: 10.2307/2002053
Stable URL: http://www.jstor.org/stable/2002053
Page Count: 14
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Topics: Polynomials, Laplace transformation, Coefficients, Numerical quadratures, Zero, Mathematical integrals, Personnel evaluation, Mathematical constants, Differential equations, Analytics
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Abstract
In finding f(t), the inverse LAPLACE transform of F(p), where (1) f(t) = (1/2π j) ∫^{c+j∞}_{cj∞} e^{pt}F(p)dp, the function F(p) may be either known only numerical or too complicated for evaluating f(t) by CAUCHY's theorem. When F(p) behaves like a polynomial without a constant term, in the variable 1/p, along (c  j∞, c + j∞), one may find f(t) numerically using new quadrature formulas (analogous to those employing the zeros of the LAGUERRE polynomials in the direct Laplace transform). Suitable choice of p_{i} yields an npoint quadrature formula that is exact when ρ_{2n} is any arbitrary polynomial of the (2n)th degree in $x \equiv 1/p$ without a constant term, namely: (2) (1/2π j) ∫^{c+j∞}_{cj∞} e^{p} ρ_{2n}(1/p)dp = ∑_{i=1}^{n} A_{i}^{(n)} ρ_{2n}(1/p_{i}). In (2), $x_i \equiv 1/p_i$ are the zeros of the orthogonal polynomials $p_n(x) \equiv 1/p_i$ are the zeros of the orthogonal polynomials $p_n(x) \equiv \Pi^n_{i=1} (x  x_i)$ where (3) (1/2π j) ∫^{c+j∞}_{cj∞} e^{p}(1/p)p_{n}(1/p)(1/p)^{idp} = 0, i = 0, 1, ⋯, n  1 and A_{i}(n) correspond to the CHRISTOFFEL numbers. The normalization $P_n(1/p) \equiv (4n  2)(4n  6) \cdots 6p_n(1/p), n \geq 2$ , produces all integral coeffients. P_{n}(1/p) is proven to be (1)^{n} e^{p}p^{nd}^{n}(e^{p}/p^{n})/dp^{n}. The normalization factor is proved, in three different ways, to be given by (4) (1/2π j) ∫^{c+j∞}_{cj∞} e^{p}(1/p)[ P_{n}(1/p) ]^{2dp} = 1/2(1)^{n}. Proofs are given for the recurrence formula (5) (2n  3)P_{n}(x) = [ (4n  2)(2n  3)x + 2 ] P_{n1}(x) + (2n  1)P_{n2}(x), for n ≥ 3, and the differential equation (6) x^{2P}_{n}"(x) + (x  1)P_{n}'(x)  n^{2P}_{n}(x) = 0. The quantities p_{i}^{(n)}, 1/p_{i}^{(n)} and A_{i}^{(n)} were computed, mostly to 6S  8S, for i = 1(1)n, n = 1(1)8.
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Mathematical Tables and Other Aids to Computation © 1955 American Mathematical Society