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Concept
Dirichlet integral
Summary
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, because the Dirichlet integral is infinite in the sense of Lebesgue integration. It is, however, finite 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.
It is a good illustration of special techniques for evaluating definite integrals. The sine integral, an antiderivative of the sinc function, is not an elementary function. However 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.
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,
Evaluating the Dirichlet integral using the Laplace transform is equivalent to calculating the same double definite integral by changing the order of integration, namely,
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.
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Related concepts (4)
Nonelementary integral
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations). A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.
Antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.