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 is not Lebesgue-integrable, and so the Dirichlet integral is undefined in the sense of Lebesgue integration. It is, however, defined in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral. This can be seen by using Dirichlet's test for improper integrals. Although the sine integral, an antiderivative of the sinc function, is not an elementary function, the value of the integral (in the Riemann or Henstock sense) can be derived using various ways, including 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. 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 Evaluating the Dirichlet integral using the Laplace transform is equivalent to calculating the same double definite integral by changing the order of integration, namely, 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 integration 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