Dirichlet Integral


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Dirichlet Integral

 

(named for P. G. L. Dirichlet), the name of several types of integrals.

(1) The integral

This Dirichlet integral is also called the Dirichlet discontinous factor and is equal to π/2 for β < a, to β/4 for β = α, and to 0 for β > α. Thus, Dirichlet integral (1) is a discontinous function of parameters α and β. Dirichlet used the integral (1) in his studies of the attraction of ellipsoids. However, this integral appeared earlier in the works of J. Fourier, S. Poisson, and A. M. Legendre.

(2) The integral

where

is the so-called Dirichlet kernel. This Dirichlet integral is equal to the nth partial sum

of a Fourier series of the function f(x). Formula (2) is one of the most important formulas in the theory of Fourier series; in particular, it enabled Dirichlet to show that the Fourier series of a function with a finite number of maxima and minima converges at every point.

(3) The integral

References in periodicals archive ?
WIRTHS, Integral means and Dirichlet integral for analytic functions, Math.
in Section 2, before we present our main results, we first establish a sufficient condition for univalence for functions in A(p) and we feel that it will be useful to present the absolute estimates for the Dirichlet integrals [DELTA](r, f) and [DELTA](r, f /z) for f [member of] [SIGMA](p) and 0 < r < p.
Criteria for univalence, Integral means and Dirichlet integral for Meromorphic functions
We have used as the velocity field of a fluid the functional form derived in Casuso (2007), obtained by studying the origin of turbulence as a consequence of a new description of the density distribution of matter as a modified discontinuous Dirichlet integral. As an interesting result we have found that this functional form for velocities is a solution to the Navier-Stokes equation when considering asymptotic behaviour, i.e.
An asymptotic solution for the Navier-Stokes equation
Topics include a review of preliminaries such as continuous and Holder continuous functions, Sobolev spaces and convex analysis; classical methods such as Euler-Lagrange equations; direct methods such as the Dirichlet integral; regularity, such as the one-dimensinal case; minimal surfaces such as in the Douglas- Courant-Tonelli method; and isoperimetric inequality.
Introduction to the calculus of variations
Sobel, Uppuluri, and Frankowski [77] tabulated, to 10D, the incomplete Dirichlet integral of Type 1:
Sobel, Uppuluri, and Frankowski [78] tabulated the incomplete Dirichlet integrals of Type 2:
A survey of tables of probability distributions
By a solution of (1.1) we understand a function u [member of] [C.sup.1](a, b) such that [rho][phi](u') [member of] [C.sup.1](a, b), the equation in (1.1) holds at every point, the boundary conditions are satisfied and the Dirichlet integral [[integral].sup.b.sub.a] [rho](t) [[absolute value of u'(t)].sup.p] dt is finite.
Half-linear Sturm-Liouville problem with weights
The modulus of a quadrilateral and Dirichlet integrals. One can express the modulus of a quadrilateral (D; [z.sub.1], [z.sub.2], [z.sub.3], [z.sub.4]) in terms of the solution of the Dirichlet-Neumann problem as follows.
The modulus of a ring domain and Dirichlet integrals. Let E and F be two disjoint compact sets in the extended complex plane C[infinity].
CONFORMAL MODULUS AND PLANAR DOMAINS WITH STRONG SINGULARITIES AND CUSPS
Japanese mathematicians Fukushima (Osaka U.), Yoichi Oshima (Kumamato U.), and Masayoshi Takeda (Tohoku U.) begin with an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms, which they explain as an axiomatic extension of the classical Dirichlet integrals in the direction of Markovian semigroups.
Dirichlet forms and symmetric Markov processes, 2d ed
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