Editing Dirichlet–Jordan test (section)

{{redirect-distinguish|Dirichlet conditions|Dirichlet boundary condition}}
In [[mathematics]], the '''Dirichlet–Jordan test''' gives [[sufficient condition]]s for a [[complex numbers|complex-valued]], [[periodic function]] <math>f</math> to be equal to the sum of its [[Fourier series]] at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). It is one of many conditions for the [[convergence of Fourier series]].

The original test was established by [[Peter Gustav Lejeune Dirichlet]] in 1829,<ref>{{citation|author=Dirichlet|year=1829|title=Sur la convergence des series trigonometriques qui servent à represénter une fonction arbitraire entre des limites donnees|journal=J. Reine Angew. Math.|volume= 4|pages=157–169}}</ref> for piecewise [[monotone function]]s (functions with a finite number of sections per period each of which is monotonic). It was extended in the late 19th century by [[Camille Jordan]] to functions of [[bounded variation]] in each period (any function of bounded variation is the difference of two monotonically increasing functions).<ref name="Fourier series and Fourier integrals"/><ref>{{citation|author=C. Jordan|title= Cours d'analyse de l'Ecole Polytechnique, t.2, calcul integral|publisher= Gauthier-Villars, Paris, 1894}}</ref>