A class of laplace inversions-evaluation of l-1[πe-pi-nn(p-2-a-2)]

The popular and powerful method of Laplace transform sometimes brings fret and frustration to the user because of difficulties occasionally encountered in the final inversion process. One such case in point is found in a viscoelastokinetic problem where formulas for the final inversion of the solutions in the transformed plane are not available in the literature. In this paper, for the purpose of bridging the gap, two Laplace inversions of fundamental importance are reported. The first, £
^-1
[πe
^-p
I
_{1(p²-a²)], is evaluated by solving an integral equation of Poisson's type on a digital computer. The results of this inversion are presented in the form of graphs for several values of the parameter a. The second inversion, £^-1[πe-pI2(p2-a-2)], is found in closed form. With these inversions and the available inversion for the zeroth order function, the inversions of any other higher order modified Bessel function with similar arguments can be obtained either by the use of the usual recurrence formula, or by a specially developed general reduction formula. The modified Bessel functions of any integral order can be reduced to terms involving only the zeroth and the first order functions.}