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

Owen T. Hanna, Richard A. Davis

Research output: Contribution to journalArticlepeer-review

## Abstract

Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: I = ∫A 0 tα exp(-xtβ)F(t)dt 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 the improved asymptotic (x → ∞) approximation of these integrals when the upper limit A equal to ∞ [the general Watson lemma (WL) problem for any A if x → ∞]. A procedure is developed here that extends application of the previous improvement methods to the difficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems having growing exponential 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 for infinite integrals (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 only F(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 at http://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 positive A values, over a very wide range in x, for various α, β, and F(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results.

Original language English (US) 485-500 16 Chemical Engineering Communications 204 4 https://doi.org/10.1080/00986445.2016.1277524 Published - Apr 3 2017

## Keywords

• Asymptotics
• Expansion-point improvement
• Exponential improvement
• Laplace Integrals with Finite Upper Limits (FUL)
• Watson’s lemma (WL)