The application of rational approximations to the solution of problems in theoretical seismology

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Abstract
The method of approximate Laplace transform inversion, by analytically inverting suitable rational approximations to a function f¯(p) of the operator p is applied to the problem of propagation from a quadratic pulse point source in a uniform liquid sphere. Apart from this main problem, the relation between the efficiency of the method and the sharpness of the pulse is illustrated in a general way by considering three simple examples (in descending order of sharpness), namely the approximate inversion of the Laplace transforms of a delayed Heaviside unit function, a simple function having a gradient discontinuity, and a quadratic pulse. In order to suggest how the technique can be extended to more difficult theoretical seismic problems, the method is applied to a simple function f¯(p) having branch points on the imaginary axis in the p-plane.

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