# Applied Laplace Transforms and z-Transforms for Scientists by Urs Graf

00). For analytical work the Post-Widder Inversion formula had little impact despite the fact that a natural probabilistic interpretation can be given to it. Today however, this inversion formula is the starting point for an important method for the numerical inversion of a Laplace transform, the Post-Widder-Stehfestalgorithm (see Chapter 11). The most important inversion formula is the Complex Inversion Formula It is based on Fourier's Integral Theorem which belongs to the realm of Fourier series and Fourier transforms.

27. By use of the Mellin transform correspondence e±it ~e±i(lll2lz f(z), O

The Bessel functions describe some damped oscillations as is shown in the graphs of Figs. 10. 9: Bessel Function of order zero. l0: Bessel Function of order 1. 2, it will be shown that for large values of t the following asymptotic expression holds: In(t) "" " {2 ( trn 7r) -;;t cos t-""2 - "4 ' t----+ 00. If n = 0, we have c = 1. Thus, lo(t) 0 1 -. 29) If n = 1, we also have c = 1. Moreover liIIlt-+o+ y(t) / t = 11 (0 +) = 0 exists, hence the image integration rule can be applied: 31 Laplace Transfonnation 11(t) =- 0_.