several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of...

Johann Peter Gustav Lejeune Dirichlet (/ˌdɪərɪˈkleɪ/; German: [ləˈʒœn diʁiˈkleː]; 13 February 1805 – 5 May 1859) was a German mathematician. In number...

Gustav Lejeune Dirichlet (1805–1859). In finite-element analysis, the essential or Dirichlet boundary condition is defined by weighted-integral form of a differential...

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

making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet. The name "Dirichlet's principle"...

Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable. The Dirichlet function is Lebesgue-integrable...

_{0}^{1}{\frac {x^{\alpha }-1}{\ln x}}dx.\end{aligned}}} The first integral, the Dirichlet integral, is absolutely convergent for positive α but only conditionally...

definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 1 where its argument...

{\sin {x}}{x}}\,dx={\frac {\pi }{2}}} (see sinc function and the Dirichlet integral) ∫ 0 ∞ sin 2 ⁡ x x 2 d x = π 2 {\displaystyle \int _{0}^{\infty }{\frac...

In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence...

In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a complex-valued, periodic function f {\displaystyle f} to be equal to the sum...

ComplexPlot3D Mathematics portal Böhmer integral Fresnel zone Track transition curve Euler spiral Zone plate Dirichlet integral Abramowitz & Stegun 1983, eqn 7...

(Fresnel integral) sin ⁡ ( x ) x = sinc ⁡ ( x ) {\displaystyle {\frac {\sin(x)}{x}}=\operatorname {sinc} (x)} (sine integral, Dirichlet integral) e − 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...

antiderivative whose value is zero at x = ∞. Their difference is given by the Dirichlet integral, 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...

In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ⁡ ( α ) {\displaystyle \operatorname...

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process...

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...

geometry, and also for his fundamental study on Dirichlet integrals, known as the Lobachevsky integral formula. William Kingdon Clifford called Lobachevsky...

dx=\operatorname {rect} (0)=1} is an improper integral (see Dirichlet integral) and not a convergent Lebesgue integral, as ∫ − ∞ ∞ | sin ⁡ ( π x ) π x | d x =...

function) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms can be defined on any...

infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes...

{\displaystyle g} that produces a third function f ∗ g {\displaystyle f*g} , as the integral of the product of the two functions after one is reflected about the y-axis...

antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function f is a differentiable function F whose...

In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions...

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...

\mathbb {C} } ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on D ( Ω )...

the Risch algorithm. other nonelementary integrals, including the Dirichlet integral and elliptic integral. It follows directly from the definition that...

in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number...