### 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) |
---|---|

Pages (from-to) | 485-500 |

Number of pages | 16 |

Journal | Chemical Engineering Communications |

Volume | 204 |

Issue number | 4 |

DOIs | |

State | Published - Apr 3 2017 |

### Keywords

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

### Cite this

**General Laplace Integral Problems : Accuracy Improvement and Extension to Finite Upper Limits.** / Hanna, Owen T.; Davis, Richard A.

Research output: Contribution to journal › Article

*Chemical Engineering Communications*, vol. 204, no. 4, pp. 485-500. https://doi.org/10.1080/00986445.2016.1277524

}

TY - JOUR

T1 - General Laplace Integral Problems

T2 - Accuracy Improvement and Extension to Finite Upper Limits

AU - Hanna, Owen T.

AU - Davis, Richard A.

PY - 2017/4/3

Y1 - 2017/4/3

N2 - 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.

AB - 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.

KW - Asymptotics

KW - Expansion-point improvement

KW - Exponential improvement

KW - Laplace Integrals with Finite Upper Limits (FUL)

KW - Watson’s lemma (WL)

UR - http://www.scopus.com/inward/record.url?scp=85014694321&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014694321&partnerID=8YFLogxK

U2 - 10.1080/00986445.2016.1277524

DO - 10.1080/00986445.2016.1277524

M3 - Article

VL - 204

SP - 485

EP - 500

JO - Chemical Engineering Communications

JF - Chemical Engineering Communications

SN - 0098-6445

IS - 4

ER -