Dirichlet integral

(Redirected from Dirichlet Integrals)

In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:

Peter Gustav Lejeune Dirichlet

This integral is not absolutely convergent, meaning has infinite Lebesgue or Riemann improper integral over the positive real line, so the sinc function is not Lebesgue integrable over the positive real line. The sinc function is, however, integrable in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral.[1][2] This can be seen by using Dirichlet's test for improper integrals.

It is a good illustration of special techniques for evaluating definite integrals, particularly when it is not useful to directly apply the fundamental theorem of calculus due to the lack of an elementary antiderivative for the integrand, as the sine integral, an antiderivative of the sinc function, is not an elementary function. In this case, the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour integration, and the Dirichlet kernel.

Evaluation

edit

Laplace transform

edit

Let   be a function defined whenever   Then its Laplace transform is given by   if the integral exists.[3]

A property of the Laplace transform useful for evaluating improper integrals is   provided   exists.

In what follows, one needs the result   which is the Laplace transform of the function   (see the section 'Differentiating under the integral sign' for a derivation) as well as a version of Abel's theorem (a consequence of the final value theorem for the Laplace transform).

Therefore,  

Double integration

edit

Evaluating the Dirichlet integral using the Laplace transform is equivalent to calculating the same double definite integral by changing the order of integration, namely,     The change of order is justified by the fact that for all  , the integral is absolutely convergent.

Differentiation under the integral sign (Feynman's trick)

edit

First rewrite the integral as a function of the additional variable   namely, the Laplace transform of   So let  

In order to evaluate the Dirichlet integral, we need to determine   The continuity of   can be justified by applying the dominated convergence theorem after integration by parts. 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 the sine function in terms of complex exponentials:  

Therefore,  

Integrating with respect to   gives  

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

Finally, by continuity at   we have   as before.

Complex contour integration

edit

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[4]  

The pole has been moved to the negative imaginary axis, so   can be integrated along the semicircle   of radius   centered at   extending in the positive imaginary direction, and closed along the real axis. One then 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   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, we get  

Finally,  

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  

Dirichlet kernel

edit

Consider the well-known formula for the Dirichlet kernel:[5] 

It immediately follows that: 

Define  

Clearly,   is continuous when   to see its continuity at 0 apply L'Hopital's Rule:  

Hence,   fulfills the requirements of the Riemann-Lebesgue Lemma. This means:  

(The form of the Riemann-Lebesgue Lemma used here is proven in the article cited.)

We would like to compute:  

However, we must justify switching the real limit in   to the integral limit in   which will follow from showing that the limit does exist.

Using integration by parts, we have:  

Now, as   and   the term on the left converges with no problem. See the list of limits of trigonometric functions. We now show that   is absolutely integrable, which implies that the limit exists.[6]

First, we seek to bound the integral near the origin. Using the Taylor-series expansion of the cosine about zero,  

Therefore,  

Splitting the integral into pieces, we have  

for some constant   This shows that the integral is absolutely integrable, which implies the original integral exists, and switching from   to   was in fact justified, and the proof is complete.

See also

edit

References

edit
  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. Archived from the original (PDF) on 18 November 2017. Retrieved 10 June 2017.
  2. ^ Bartle, Robert G.; Sherbert, Donald R. (2011). "Chapter 10: The Generalized Riemann Integral". Introduction to Real Analysis. John Wiley & Sons. pp. 311. ISBN 978-0-471-43331-6.
  3. ^ Zill, Dennis G.; Wright, Warren S. (2013). "Chapter 7: The Laplace Transform". Differential Equations with Boundary-Value Problems. Cengage Learning. pp. 274-5. ISBN 978-1-111-82706-9.
  4. ^ Appel, Walter. Mathematics for Physics and Physicists. Princeton University Press, 2007, p. 226. ISBN 978-0-691-13102-3.
  5. ^ Chen, Guo (26 June 2009). A Treatment of the Dirichlet Integral Via the Methods of Real Analysis (PDF) (Report).
  6. ^ R.C. Daileda. Improper Integrals (PDF) (Report).
edit