Dirichlet integral explained
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:
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
Laplace transform
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 isprovided
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
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)
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
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 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
2I+\Im(ln0-ln(\pii))=2I-\pi
(here
is any branch of logarithm on upper half-plane), leading to
Dirichlet kernel
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,
= e^
.
Splitting the integral into pieces, we have
dx + \int_^ \fracdx\leq K,
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
Notes and References
- Bartle . Robert G. . Robert G. Bartle . 10 June 1996 . Return to the Riemann Integral . The American Mathematical Monthly . 103 . 8 . 625–632 . 10.2307/2974874 . 2974874 . 10 June 2017 . 18 November 2017 . https://web.archive.org/web/20171118184849/http://math.tut.fi/courses/73129/Bartle.pdf . dead .
- Book: Bartle, Robert G.. Introduction to Real Analysis. limited. Sherbert. Donald R.. John Wiley & Sons. 2011. 978-0-471-43331-6. 311. Chapter 10: The Generalized Riemann Integral.
- Book: Zill, Dennis G. . Differential Equations with Boundary-Value Problems . limited. Wright. Warren S. . Cengage Learning . 2013 . 978-1-111-82706-9. 274-5 . Chapter 7: The Laplace Transform.
- Appel, Walter. Mathematics for Physics and Physicists. Princeton University Press, 2007, p. 226. .
- A Treatment of the Dirichlet Integral Via the Methods of Real Analysis . Chen, Guo . 26 June 2009.
- Improper Integrals . R.C. Daileda.