Laplace integrals

The improper integrals


where a is a positive , are called Laplace integrals.  Both of them have the same value πe-a.

The evaluation of the Laplace integrals can be performed by first determining the integrals


where one integrates along the real axisMathworldPlanetmath.  Therefore one has to determine the integrals


around the perimeter of the half-disk with the arc in the upper half-plane, centered in the origin and with the diameter  (-R,+R).  The residue theoremMathworldPlanetmath yields the values

eizz-ia𝑑z= 2iπe-aandeizz+ia𝑑z= 0.

As in the entry example of using residue theorem, the parts of these contour integrals along the half-circle tend to zero when  R.  Consequently,

-eixx-ia𝑑x= 2iπe-aand-eixx+ia𝑑x= 0.

These equations imply by adding and subtracting and then taking the real ( and the imaginary partsDlmfMathworld, the



  • 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava. Helsinki (1963).
Title Laplace integrals
Canonical name LaplaceIntegrals
Date of creation 2013-03-22 18:43:17
Last modified on 2013-03-22 18:43:17
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Definition
Classification msc 40A10