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What are the most important applications of improper integrals, in particular to computer science and related fields, and to technology and engineering in general?

I know that improper integrals are very common in probability and statistics; also, the Laplace transform, the Fourier transform and many special functions like Beta and Gamma are defined using improper integrals, which appear in a lot of problems and computations. But what about their direct, practical applications in real life situations?

Any insight is much appreciated!

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  • $\begingroup$ Check this problem which comes from "communication system performance analysis". $\endgroup$ – Mhenni Benghorbal May 4 '14 at 20:27
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One easy example on the field of physics are those problems related to finding the electrical / graviational /etc. potential of a given field. For example, the electric potential created by a charged sphere of radius $R$ for $r \geq R$ is given by:

$$ V(r) = - \int_{\infty}^r E(r) \, \mathrm{d}r$$ where $E$ is the electric field (modulus) generated by the sphere and $r$ is the distance from the center of the sphere. This also has a very simple physical meaning since the integral (when the minus sign is considered) represents the work to be done by somebody (or something) to bring the considered amount of electrical charge (responsible for creating the electric field $E$) from $\infty$ to $r$.

The same expression and physical meaning can be applied to the case of gravitational potential.

Cheers!

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Whenever I think of improper integrals, and their applications my mind remembers all the physics based equations. Given position at all points, we can use an indefinite integral to find the speed and acceleration at all points. We can apply this rule of rates to pretty much anything to find out total work, or total volume of anything. Additionally a lot of the more intricate integrals are applicable to electromagnetic fields and how they interact with other things, and their more easily formula conforming properties.

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