Laplace integrals
The improper integrals
โซโ-โacosxx2+a2๐xโandโโซโ-โxsinxx2+a2๐x, |
where a is a positive , are called Laplace integrals.โ Both of them have the same value ฯe-a.
The evaluation of the Laplace integrals can be performed by first determining the integrals
โซโ-โeixx-ia๐xโandโโซโ-โeixx+ia๐x |
where one integrates along the real axis.โ Therefore one has to determine the integrals
โฎeizz-ia๐zโandโโฎeizz+ia๐z |
around the perimeter of the half-disk with the arc in the upper half-plane, centered in the origin and with the diameter โ(-R,+R).โ The residue theorem yields the values
โฎeizz-ia๐z=โ2iฯe-aโandโโฎeizz+ia๐z=โ0. |
As in the entry example of using residue theorem, the parts of these contour integrals along the half-circle tend to zero whenโ Rโโ.โ Consequently,
โซโ-โeixx-ia๐x=โ2iฯe-aโandโโซโ-โeixx+ia๐x=โ0. |
These equations imply by adding and subtracting and then taking the real (http://planetmath.org/RealPart) and the imaginary parts, the
โซโ-โacosxx2+a2๐x=โซโ-โxsinxx2+a2๐x=ฯe-a. |
References
- 1 R. Nevanlinna & V. Paatero: Funktioteoria.โ Kustannusosakeyhtiรถ Otava. Helsinki (1963).
Title | Laplace integrals |
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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 |