General Laplace Integral Problems: Accuracy Improvement and Extension to Finite Upper Limits.

Many problems in engineering and science involve calculation of difficult Laplace integrals of the form:In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for theimprovedasymptotic (x→ ∞) approximation of these integralswhen the upper limit A equal to ∞[the general Watson lemma (WL) problem for anyAifx→ ∞]. A procedure is developed here that extends application of the previous improvement methods to thedifficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems havinggrowingexponential behavior, certain infinite Fourier integrals and problems having large (tα) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed forinfiniteintegrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new “generalized incomplete gamma function (IGF)” algorithm which itself utilizes an improvement procedure. The new FUL procedure requires onlyF(t),F′(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users athttp://www.d.umn.edu/~rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positiveAvalues, over a very wide range inx, for variousα, β,andF(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results. [ABSTRACT FROM AUTHOR]