Improper Integrals

What is an improper integral? An improper integral is the one which has the definite integral that has either/both the limits infinite or an integrand that goes to infinity at single or more points in the integration range.

It is much required to make use of improper integrals to calculate a value for integrals that may not exist in the Riemann integral form. It may be because of a uniqueness in the function, or a non - finite termination of the orbit or domain of integration.

Convergence of Improper integrals:

An improper integral converges if the integrals under the limit exist for all sufficiently large value.

Divergence of Improper integrals:

An improper integral can also diverge to infinity if the value of infinity (or - infinity) is assigned to the integral. However, integrals involving trigonometric functions might not diverge in a particular direction as it does not exist even as an extended real number.

Limitations of Improper Integrals:

The limit should be taken with respect to one endpoint at a time is a limitation of the improper integration technique. Sometimes, it is also possible to define improper integrals where both endpoints are infinite, i.e., Gaussian integral. But it is not possible to define other integrals unambiguously if the double limit diverges. However, in the case of Cauchy principal value, an improper integral can be defined.

Improper Lebesgue integrals:

In a Lebesgue integral, the Riemann integral can be defined without reference to the limit but could not alternatively be easily calculated. However, in some cases, where the integral is not even defined due to the positive and negative parts of the function are infinite, but the limits exist, then they are referred to as properly improper integral.

Uniqueness:

The uniqueness/singularities of an improper integral i e., the points of the extended real number line at which limits are used could be discussed elaborately. This integral is usually symbolically written as a standard definite integral, might be with infinity as the limit of integration. But it hides the limiting process. Instead of using the Riemann Integral, in some cases bypass this requirement by using advanced Lebesgue integral. But, to evaluate limit to definite result, this may not help. It is required in the theoretical treatment for the usage of Fourier transform along with spreading use of integrals over the entire real line.