In calculus, we define an improper (real) integral with two infinite limits as follows:

We can compute such integrals using integrals of functions of a complex variable. Consider the loop consisting of the segment on the axis and the upper semicircle defined by (with ),as in Figure 1, choosing such that this loop encloses all the singular points of the complex function which are in the upper half-plane.

Then we compute the integral

We can show that

thus

Let . This function has 4 poles in the complex plane, namely the solutions of the equation . These poles are:

Thus we have:

Let be a positive real number such that all of these poles with positive imaginary part are inside the loop defined as above, namely the first and the fourth of the numbers listed in 6; these are

and | (9.3) |

Now we compute the residues of at each of these points; in what follows, and are simple loops obtained by ``cutting'' into two parts, such that is an interior point of , but not of , and is an interior point of , but not of (cf Figure 2).

Res | ||

Res | ||

Res | Res |

Thus, we have:

Noah Dana-Picard 2007-12-24