## Abstract

Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: I = ∫^{A}_{0} t^{α} exp(-xt^{β})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 (t^{α}) 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) |
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Pages (from-to) | 485-500 |

Number of pages | 16 |

Journal | Chemical Engineering Communications |

Volume | 204 |

Issue number | 4 |

DOIs | |

State | Published - Apr 3 2017 |

### Bibliographical note

Publisher Copyright:© 2017, Copyright © Taylor & Francis Group, LLC.

## Keywords

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