improper integral

An improper integral occurs when one or more of these conditions is not satisfied.

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

It is also possible for an improper integral to diverge to infinity.

An improper integral converges if the limit defining it exists.

In using improper integrals, it can matter which integration theory is in play.

It can be expressed as an application of a Cauchy principal value improper integral.

The gamma function is an example of a special function, defined as an improper integral.

In this case, there are more sophisticated definitions of the limit which can produce a convergent value for the improper integral.

Comparing these two computations yields the integral, though one should take care about the improper integrals involved.

Thus this is a doubly improper integral.

The improper integral is often extended to negative values of x via the conventional choice:

It is possible to prove that the harmonic series diverges by comparing its sum with an improper integral.

An integral is (C, 0) summable precisely when it exists as an improper integral.

For complex numbers with a positive real part, it is defined via an improper integral that converges:

It can also be defined as a pair of distinct improper integrals of the first kind:

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

For example, the value can be determined from attempts to evaluate a double improper integral, or by using differentiation under the integral sign.

This improper integral is convergent.

An example of an improper integrals where both endpoints are infinite is the Gaussian integral .

Thus improper integrals are clearly useful tools for obtaining the actual values of integrals.

The simplest possible extension is to define such an integral as a limit, in other words, as an improper integral.

One can speak of the singularities of an improper integral, meaning those points of the extended real number line at which limits are used.

A semi-infinite integral is an improper integral over a semi-infinite interval.

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions.

If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity.