The improper integrals
where is a positive , are called Laplace integrals. Both of them have the same value .
The evaluation of the Laplace integrals can be performed by first determining the integrals
As in the entry example of using residue theorem, the parts of these contour integrals along the half-circle tend to zero when . Consequently,
These equations imply by adding and subtracting and then taking the real (http://planetmath.org/RealPart) and the imaginary parts, the
- 1 R. Nevanlinna & V. Paatero: Funktioteoria. Kustannusosakeyhtiö Otava. Helsinki (1963).