Lobachevsky integral formula

Last updated September 25, 2023

In mathematics, Dirichlet integrals play an important role in distribution theory. We can see the Dirichlet integral in terms of distributions.

One of those is the improper integral of the sinc function over the positive real line,

\int _{0}^{\infty }{\frac {\sin x}{x}}\,dx=\int _{0}^{\infty }{\frac {\sin ^{2}x}{x^{2}}}\,dx={\frac {\pi }{2}}.

Lobachevsky's Dirichlet integral formula

Let $f(x)$ be a continuous function satisfying the $\pi$ -periodic assumption $f(x+\pi )=f(x)$ , and $f(\pi -x)=f(x)$ , for $0\leq x<\infty$ . If the integral $\int _{0}^{\infty }{\frac {\sin x}{x}}f(x)\,dx$ is taken to be an improper Riemann integral, we have Lobachevsky's Dirichlet integral formula

\int _{0}^{\infty }{\frac {\sin ^{2}x}{x^{2}}}f(x)\,dx=\int _{0}^{\infty }{\frac {\sin x}{x}}f(x)\,dx=\int _{0}^{\pi /2}f(x)\,dx

Moreover, we have the following identity as an extension of the Lobachevsky Dirichlet integral formula^[1]

\int _{0}^{\infty }{\frac {\sin ^{4}x}{x^{4}}}f(x)\,dx=\int _{0}^{\pi /2}f(t)\,dt-{\frac {2}{3}}\int _{0}^{\pi /2}\sin ^{2}tf(t)\,dt.

As an application, take $f(x)=1$ . Then

\int _{0}^{\infty }{\frac {\sin ^{4}x}{x^{4}}}\,dx={\frac {\pi }{3}}.

Related Research Articles

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer $n$ ,

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

The Fresnel integrals $S (x)$ and $C (x)$ are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function ( $erf$ ). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

<span class="mw-page-title-main">Improper integral</span> Concept in mathematical analysis

In mathematical analysis, an 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 of Riemann integrals, this typically involves unboundedness, either of the set over which the integral is taken or of the integrand, or both. It may also involve bounded but not closed sets or bounded but not continuous functions. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents a limit of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge. If a regular definite integral is worked out as if it is improper, the same answer will result.

In mathematics, the Clausen function, introduced by Thomas Clausen (1832), is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.

In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function.

In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0:

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function $over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is$

In mathematics, physics and engineering, the sinc function, denoted by $sinc(x)$ , has two forms, normalized and unnormalized.

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

<span class="mw-page-title-main">Dirichlet integral</span> Integral of sin(x)/x from 0 to infinity.

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:

In mathematics, the Leibniz formula for $π$ , named after Gottfried Wilhelm Leibniz, states that

In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form

In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.

<span class="mw-page-title-main">Multiple integral</span> Generalization of definite integrals to functions of multiple variables

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, $f (x, y)$ or $f (x, y, z)$ . Integrals of a function of two variables over a region in $(the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals . For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.$

In calculus, integration by parametric derivatives, also called parametric integration, is a method which uses known Integrals to integrate derived functions. It is often used in Physics, and is similar to integration by substitution.

In mathematics, a Borwein integral is an integral whose unusual properties were first presented by mathematicians David Borwein and Jonathan Borwein in 2001. Borwein integrals involve products of $, where the sinc function is given by for not equal to 0, and .$

In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that

In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral transform that inverts the operation of forming the DGF of a sequence. This inversion is analogous to performing an inverse Z-transform to the generating function of a sequence to express formulas for the series coefficients of a given ordinary generating function.

References

↑ Jolany, Hassan (2018). "An extension of Lobachevsky formula". Elemente der Mathematik. 73: 89–94.

Hardy, G. H., The Integral $\int _{0}^{\infty }{\frac {\sin x}{x}}\,dx={\frac {\pi }{2}},$ The Mathematical Gazette, Vol. 5, No. 80 (June–July 1909), pp. 98–103 JSTOR 3602798
Dixon, A. C., Proof That $\int _{0}^{\infty }{\frac {\sin x}{x}}\,dx={\frac {\pi }{2}},$ The Mathematical Gazette, Vol. 6, No. 96 (January 1912), pp. 223–224. JSTOR 3604314

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Jolany, Hassan (2018). "An extension of Lobachevsky formula". Elemente der Mathematik. 73: 89–94.

[1]