Search results
Results from the WOW.Com Content Network
The Dirichlet function is not Riemann-integrable on any segment of despite being bounded because the set of its discontinuity points is not negligible (for the Lebesgue measure). The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral.
The Dirichlet L-function L(s, χ) = 1 − 3 −s + 5 −s − 7 −s + ⋅⋅⋅ (sometimes given the special name Dirichlet beta function), with trivial zeros at the negative odd integers. Let χ be a primitive character modulo q, with q > 1. There are no zeros of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative ...
The convolution of D n (x) with any function f of period 2 π is the nth-degree Fourier series approximation to f, i.e., we have () = () = = ^ (), where ^ = is the k th Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
The name "Dirichlet's principle" is due to Bernhard Riemann, who applied it in the study of complex analytic functions. [1]Riemann (and others such as Carl Friedrich Gauss and Peter Gustav Lejeune Dirichlet) knew that Dirichlet's integral is bounded below, which establishes the existence of an infimum; however, he took for granted the existence of a function that attains the minimum.
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...
Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. E[u] ≥ 0 for every function u. Solving Laplace's equation () = for all , subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function u that satisfies the boundary conditions and has minimal Dirichlet energy.
The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +:
Dirichlet also studied the first boundary-value problem, for the Laplace equation, proving the uniqueness of the solution; this type of problem in the theory of partial differential equations was later named the Dirichlet problem after him. A function satisfying a partial differential equation subject to the Dirichlet boundary conditions must ...