Improper Integral - Convergence of The Integral

Convergence of The Integral

An improper integral converges if the limit defining it exists. Thus for example one says that the improper integral

exists and is equal to L if the integrals under the limit exist for all sufficiently large t, and the value of the limit is equal to L.

It is also possible for an improper integral to diverge to infinity. In that case, one may assign the value of ∞ (or −∞) to the integral. For instance

However, other improper integrals may simply diverge in no particular direction, such as

which does not exist, even as an extended real number.

A limitation of the technique of improper integration is that the limit must be taken with respect to one endpoint at a time. Thus, for instance, an improper integral of the form

is defined by taking two separate limits; to wit

provided the double limit is finite. By the properties of the integral, this can also be written as a pair of distinct improper integrals of the first kind:

where c is any convenient point at which to start the integration.

It is sometimes possible to define improper integrals where both endpoints are infinite, such as the Gaussian integral . But one cannot even define other integrals of this kind unambiguously, such as, since the double limit diverges:

In this case, one can however define an improper integral in the sense of Cauchy principal value:

The questions one must address in determining an improper integral are:

• Does the limit exist?
• Can the limit be computed?

The first question is an issue of mathematical analysis. The second one can be addressed by calculus techniques, but also in some cases by contour integration, Fourier transforms and other more advanced methods.