Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms
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- by Herbert E. Salzer PDF
- Math. Comp. 9 (1955), 164-177 Request permission
Abstract:
In finding $f(t)$, the inverse Laplace transform of $F(p)$, where (1) $f(t) = (1/2\pi j)\int _{c - j\infty }^{c + j\infty }{e^{{p^t}}}F(p)dp$, the function $F(p)$ may be either known only numerically 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 ($c - j\infty ,\;c + j\infty$), 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 $n$-point quadrature formula that is exact when ${\rho _{2n}}$ is any arbitrary polynomial of the ($2n$)th degree in $x \equiv 1/p$ without a constant term, namely: (2) $(1/2\pi j)\int _{c - j\infty }^{c + j\infty } {{e^p}} {\rho _{2n}}(1/p)dp = {\sum \limits _{i = 1}^n {{A_i}} ^{(n)}}{\rho _{2n}}(1/{p_i})$. In (2), ${x_i} \equiv 1/{p_i}$ are the zeros of the orthogonal polynomials ${P_n}(x) \equiv \prod \limits _{i = 1}^n {(x - {x_i})}$ where (3) $(1/2\pi j)\int _{c - j\infty }^{c + j\infty } {{e^p}} (1/p){p_n}(1/p){(1/p)^i}dp = 0,\;i = 0,1, \cdots ,n - 1$ and ${A_i}^{(n)}$ correspond to the Christoffel numbers. The normalization ${P_n}(1/p) \equiv (4n - 2)(4n - 6) \cdots 6{p_n}(1/p),n \geq 2$, produces all integral coefficients. ${P_n}(1/p)$ is proven to be ${( - 1)^n}{e^{ - p}}{p^n}{d^n}({e^p}/{p^n})/d{p^n}$. The normalization factor is proved, in three different ways, to be given by (4) $(1/2\pi j)\int _{c - j\infty }^{c + j\infty } {{e^p}} (1/p){[{P_n}(1/p)]^2}dp = \tfrac {1}{2}{( - 1)^n}$. Proofs are given for the recurrence formula (5) $(2n - 3){P_n}(x) = [(4n - 2)(2n - 3)x + 2]{P_{n - 1}}(x) + (2n - 1){P_{n - 2}}(x)$, for $n \geq 3$, and the differential equation (6) ${x^2}{P_n}^{”}(x) + (x - 1){P_n}’(x) - {n^2}{P_n}(x) = 0$. The quantities ${p_i}^{(n)},1/{p_i}^{(n)}$ and ${A_i}^{(n)}$ were computed, mostly to 6S – 8S, for $i = 1(1)n,n = 1(1)8$.References
- H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Oxford University Press, New York, 1941. MR 0005988
- Herbert E. Salzer and Ruth Zucker, Table of the zeros and weight factors of the first fifteen Laguerre polynomials, Bull. Amer. Math. Soc. 55 (1949), 1004–1012. MR 32191, DOI 10.1090/S0002-9904-1949-09327-8 G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Colloquium Pub., v. 23, 1939, p. 46-47.
- H. L. Krall and Orrin Frink, A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), 100–115. MR 28473, DOI 10.1090/S0002-9947-1949-0028473-1 G. Doetsch, Theorie und Anwendung der Laplace-Transformation, Springer, Berlin, 1937, p. 128. The shift in notation from $(n + 1)$ to $n$ in ${A_i}^{(n)}$ will cause no confusion after the ${A_i}$’s have been computed and are ready for use in (6). It was called to the author’s attention by H. L. Krall that ${P_n}(x) \equiv {( - 1)^n}{y_n}(x,1, - 1)$ where ${y_n}(x,a,b)$ are “generalized Bessel polynomials” (see[4]). G. Szegö, op. cit., p. 41-42. Formula (14) holds for $n = 2$ if we define ${P_0}(x) \equiv 1$. G. Szegö, op. cit., p. 341-342.
Additional Information
- © Copyright 1955 American Mathematical Society
- Journal: Math. Comp. 9 (1955), 164-177
- MSC: Primary 42.1X
- DOI: https://doi.org/10.1090/S0025-5718-1955-0078498-1
- MathSciNet review: 0078498