In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, there are several integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

s known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet.

One of those is

This can be derived from attempts to evaluate a double improper integral

Improper integral

In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....

 two different ways. It can also be derived using differentiation under the integral sign.

Double Improper Integral Method

Pre-knowledge of properties of Laplace transforms allows us to evaluate this Dirichlet integral succinctly in the following manner:

This is equivalent to attempting to evaluate the same double definite integral in two different ways, by reversal of the order of integration

Order of integration (calculus)

In calculus, interchange of the order of integration is a methodology that transforms iterated integrals of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed...

, viz.,

Differentiation under the integral sign

First rewrite the integral as a function of variable . Let

then we need to find .

Differentiate with respect to and apply the Leibniz Integral Rule

Leibniz integral rule

In mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the formthen for x \in the derivative of this integral is thus expressible...

 to obtain:

This integral was evaluated without proof, above, based on Laplace trasform tables; we derive it this time. It is made much simpler by recalling Euler's formula,

then,

where represents the imaginary part.

Integrating with respect to :

where is a constant to be determined. As,

for some integers m & n. It is easy to show that has to be zero, by analyzing easily observed bounds for this integral:

End of proof.

Extending this result further, with the introduction of another variable, first noting that is an even function and therefore

then:

Complex integration

The same result can be obtained via complex integration. Let's consider

As a function of the complex variable z, it has a simple pole at the origin, which prevents the application of Jordan's lemma, whose other hypotheses are satisfied. We shall then define a new function g(z) as follows

The pole has been moved away from the real axis, so g(z) can be integrated along the semicircle of radius R centered at z=0 and closed on the real axis, then the limit should be taken.

The complex integral is zero by the residue theorem, as there are no poles inside the integration path

The second therm vanishes as R goes to infinity; for arbitrarily small , the Sokhatsky–Weierstrass theorem applied to the first one yelds

Where P.V. indicates Cauchy Principal Value

Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...

. By taking the imaginary part on both sides and noting that is even and by definition , we get the desired result

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.