Dirichlet integral

In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet.

One of those is the improper integral of the sinc function over the positive real line:

This integral is not absolutely convergent, and so the integral is not even defined in the sense of Lebesgue integration, but it is defined in the sense of the improper Riemann integral or the Henstock–Kurzweil integral.[1] The value of the integral (in the Riemann or Henstock sense) can be derived in various ways. For example, the value can be determined from attempts to evaluate a double improper integral, or by using differentiation under the integral sign.

EvaluationEdit

Double improper integral methodEdit

One of the well-known properties of Laplace transforms is

 

which allows one to evaluate the Dirichlet integral succinctly in the following manner:

 

because   is the Laplace transform of the function  . This is equivalent to attempting to evaluate the same double definite integral in two different ways, by reversal of the order of integration, namely:

 
 

Differentiation under the integral sign (Feynman's trick)Edit

First rewrite the integral as a function of the additional variable  . Let

 

In order to evaluate the Dirichlet integral, we need to determine .

Differentiate with respect to   and apply the Leibniz rule for differentiating under the integral sign to obtain

 

Now, using Euler's formula   one can express a sinusoid in terms of complex exponential functions. We thus have

 

Therefore,

 

Integrating with respect to   gives

 

where   is a constant of integration to be determined. Since     using the principal value. This means

 

Finally, for  , we have  , as before.

Complex integrationEdit

The same result can be obtained by complex integration. Consider

 

As a function of the complex variable  , it has a simple pole at the origin, which prevents the application of Jordan's lemma, whose other hypotheses are satisfied.

Define then a new function[2]

 

The pole has been moved away from the real axis, so   can be integrated along the semicircle of radius   centered at   and closed on the real axis. One than takes the limit  .

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

 

The second term vanishes as R goes to infinity. As for the first integral, one can use one version of the Sokhotski–Plemelj theorem for integrals over the real line: for a complex-valued function f defined and continuously differentiable on the real line and real constants   and   with   one finds

 

where   denotes the Cauchy principal value. Back to the above original calculation, one can write

 

By taking the imaginary part on both sides and noting that the function   is even, so

 

the desired result is obtained as

 

Alternatively, choose as the integration contour for   the union of upper half-plane semicircles of radii   and   together with two segments of the real line that connect them. On one hand the contour integral is zero, independently of   and  ; on the other hand, as   and   the integral's imaginary part converges to   (here   is any branch of logarithm on upper half-plane), leading to  .

Via the Dirichlet kernelEdit

Let

 

be the Dirichlet kernel.

This is clearly symmetric about zero, that is,

 

for all x, and

 

since   for any  .

Define

 

This is continuous on the interval  , so it is bounded by     for some constant , and hence by the Riemann–Lebesgue lemma,

 

Therefore,

 

by the above.

See alsoEdit

NotesEdit

  1. ^ Bartle, Robert G. (10 June 1996). "Return to the Riemann Integral" (PDF). The American Mathematical Monthly. 103 (8): 625–632. doi:10.2307/2974874. JSTOR 2974874.
  2. ^ Appel, Walter. Mathematics for Physics and Physicists. Princeton University Press, 2007, p. 226. ISBN 978-0-691-13102-3.

External linksEdit