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.
^Bartle, Robert G. (10 June 1996). "Return to the Riemann Integral" (PDF). The American Mathematical Monthly. 103 (8): 625–632. doi:10.2307/2974874. JSTOR 2974874. Archived from the original (PDF) on 18 November 2017. Retrieved 10 June 2017.
^Bartle, Robert G.; Sherbert, Donald R. (2011). "Chapter 10: The Generalized Riemann Integral". Introduction to Real Analysis. John Wiley & Sons. pp. 311. ISBN 978-0-471-43331-6.
and 27 Related for: Dirichlet integral information
several integrals known as the Dirichletintegral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of...
Johann Peter Gustav Lejeune Dirichlet (German: [ləˈʒœn diʁiˈkleː]; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved...
In mathematics, Dirichletintegrals play an important role in distribution theory. We can see the Dirichletintegral in terms of distributions. One of...
Gustav Lejeune Dirichlet (1805–1859). In finite-element analysis, the essential or Dirichlet boundary condition is defined by weighted-integral form of a differential...
_{0}^{1}{\frac {x^{\alpha }-1}{\ln x}}dx.\end{aligned}}} The first integral, the Dirichletintegral, is absolutely convergent for positive α but only conditionally...
Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable. The Dirichlet function is Lebesgue-integrable...
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process...
(Fresnel integral) sin ( x ) x = sinc ( x ) {\displaystyle {\frac {\sin(x)}{x}}=\operatorname {sinc} (x)} (sine integral, Dirichletintegral) e − x...
definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 1 where its argument...
antiderivative whose value is zero at x = ∞. Their difference is given by the Dirichletintegral, Si ( x ) − si ( x ) = ∫ 0 ∞ sin t t d t = π 2 or Si ( x...
In mathematics, a Dirichlet series is any series of the form ∑ n = 1 ∞ a n n s , {\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},} where s...
{\sin {x}}{x}}\,dx={\frac {\pi }{2}}} (see sinc function and the Dirichletintegral) ∫ 0 ∞ sin 2 x x 2 d x = π 2 {\displaystyle \int _{0}^{\infty }{\frac...
dx=\operatorname {rect} (0)=1} is an improper integral (see Dirichletintegral) and not a convergent Lebesgue integral, as ∫ − ∞ ∞ | sin ( π x ) π x | d x =...
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region...
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ( α ) {\displaystyle \operatorname...
: 13–15 Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered. The first integral, with broad...
mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear...
mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x...
improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context...
the Risch algorithm. other Nonelementary integrals, including the Dirichletintegral and elliptic integral. It follows directly from the definition that...
\mathbb {C} } ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichletintegral defines a seminorm on D ( Ω )...
antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative...
}}p>0\\\\0&{\text{if }}p=0\\\\-{\dfrac {\pi }{2}}&{\text{if }}p<0\end{cases}}} (see Dirichletintegral) ∫ 0 ∞ sin p x cos q x x d x = { 0 if q > p > 0 π 2 if 0...
function) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms can be defined on any...
gradient vector field of the function u. Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. E[u] ≥ 0 for every...
Global Information
