Solving integral equations by $L$ and $L^{-1}$ operators
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- by Charles Fox PDF
- Proc. Amer. Math. Soc. 29 (1971), 299-306 Request permission
Abstract:
\begin{equation}\tag {$1$}g(u) = \int _0^\infty {k(ux)f(x)dx = \frac {1}{{2\pi i}}\int _C^{} {K(s)F(1 - s){u^{ - s}}ds,} } \end{equation} where $g(u)$ and $k(u)$ are known and $f(x)$ is to be found. $K(s)$ is the Mellin transform of $k(x)$ and $F(s)$ of $f(x)$; hence the second equality. L and ${L^{ - 1}}$ denote the Laplace transform and its inverse. If \begin{equation}\tag {$2$}{{K(s) = \prod \limits _{i = 1}^n {\Gamma ({\alpha _i}s + {\beta _i})} } {\prod \limits _{j = 1}^m {\Gamma ({\alpha _j}s + {\beta _j})} }}\end{equation} then I show that a suitable combination of L and ${L^{ - 1}}$ operators, applied to (1), can eliminate $K(s)$ from the second integrand. This leaves $F(1 - s)$ standing free and the Mellin transform then obtains $f(x)$ from $F(1 - s)$. This solution needs tables of Laplace transforms only. When (2) does not hold, an L and ${L^{ - 1}}$ combination may turn (1) into an integral equation whose solution is already known.References
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Tables of integral transforms. Vol. 1, McGraw-Hill, New York, 1954.
Tables of integral transforms. Vol. 2, McGraw-Hill, New York, 1954.
- A. Erdélyi, On some functional transformations, Univ. e Politec. Torino Rend. Sem. Mat. 10 (1951), 217–234. MR 47818 E. C. Titchmarsh, Theory of Fourier integrals, Clarendon Press, Oxford, 1937. F. Tricomi, Sulla transformazione e il teorem di reciprocita de Hankel, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (6) 22 (1935), 564-571. G. N. Watson, Theory of Bessel functions, Cambridge Univ. Press, New York, 1922. E. T. Whittaker and G. N. Watson, Modern analysis, Cambridge Univ. Press, New York, 1915.
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 299-306
- MSC: Primary 44.28; Secondary 45.00
- DOI: https://doi.org/10.1090/S0002-9939-1971-0280944-6
- MathSciNet review: 0280944