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In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test .
The absolute value of in the conditions above can be replaced by either the positive or the negative part of ; these forms include Tonelli's theorem as a special case as the negative part of a non-negative function is zero and so has finite integral. Informally, all these conditions say that the double integral of is well defined, though ...
If is a compact, self-adjoint operator on the space [,] along with the relevant boundary conditions, then by the Spectral theorem there exists a basis for [,] consisting of eigenfunctions for . Let the spectrum of T {\displaystyle T} be E {\displaystyle E} and let f λ {\displaystyle f_{\lambda }} be an eigenfunction with eigenvalue λ ∈ E ...
In the study of ordinary differential equations and their associated boundary value problems in mathematics, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bernard Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see (ε, δ)-definition of limit below) to define continuous functions.
The function () is defined on the interval [,].For a given , the difference () takes the maximum at ′.Thus, the Legendre transformation of () is () = ′ (′).. In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, [1] is an involutive transformation on real-valued functions that are ...
A geometrical arrangement used in deriving the Kirchhoff's diffraction formula. The area designated by A 1 is the aperture (opening), the areas marked by A 2 are opaque areas, and A 3 is the hemisphere as a part of the closed integral surface (consisted of the areas A 1, A 2, and A 3) for the Kirchhoff's integral theorem.
Compared to the classical counterpart, one is generally less likely to prove the existence of relations that cannot be realized. A restriction to the constructive reading of existence apriori leads to stricter requirements regarding which characterizations of a set involving unbounded collections constitute a (mathematical, and so always meaning total) function.