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 . 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.

One of the well-known properties of Laplace transforms is

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

where ${\displaystyle {\mathcal {L}}\{\sin t\}(s)=1/(s^{2}+1)}$ is the Laplace transform of the function sint. This is equivalent to attempting to evaluate the same double definite integral in two different ways, by reversal of the order of integration, viz.,

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