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Beta function
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{{Short description|Mathematical function}} {{About|the Euler beta function}} [[File:Beta function.svg|thumb|[[Contour plot]] of the beta function]] In [[mathematics]], the '''beta function''', also called the [[Euler integral (disambiguation)|Euler integral]] of the first kind, is a [[special function]] that is closely related to the [[gamma function]] and to [[binomial coefficient]]s. It is defined by the [[integral]] :<math> \Beta(z_1,z_2) = \int_0^1 t^{z_1-1}(1-t)^{z_2-1}\,dt</math> for [[complex number]] inputs <math> z_1, z_2 </math> such that <math> \operatorname{Re}(z_1), \operatorname{Re}(z_2)>0</math>. The beta function was studied by [[Leonhard Euler]] and [[Adrien-Marie Legendre]] and was given its name by [[Jacques Philippe Marie Binet|Jacques Binet]]; its symbol {{math|Ξ}} is a [[Greek alphabet|Greek]] capital [[Beta (letter)|beta]].
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