Dirichlet integral

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 number line.

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.

[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.

But since the integrand is an even function, the domain of integration can be extended to the negative real number line as well.

[3] A property of the Laplace transform useful for evaluating improper integrals is

(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).

First rewrite the integral as a function of the additional variable

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

can be justified by applying the dominated convergence theorem after integration by parts.

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

one can express the sine function in terms of complex exponentials:

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

The pole has been moved to the negative imaginary axis, so

extending in the positive imaginary direction, and closed along the real axis.

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

Back to the above original calculation, one can write

By taking the imaginary part on both sides and noting that the function

Alternatively, choose as the integration contour for

the union of upper half-plane semicircles of radii

On one hand the contour integral is zero, independently of

the integral's imaginary part converges to

is any branch of logarithm on upper half-plane), leading to

to see its continuity at 0 apply L'Hopital's Rule:

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

However, we must justify switching the real limit in

the term on the left converges with no problem.

See the list of limits of trigonometric functions.

is absolutely integrable, which implies that the limit exists.

[6] First, we seek to bound the integral near the origin.