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Beta function
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== Properties == The beta function is [[symmetric function|symmetric]], meaning that <math> \Beta(z_1,z_2) = \Beta(z_2,z_1)</math> for all inputs <math>z_1</math> and <math>z_2</math>.<ref name=Davis622>{{citation | last = Davis | first = Philip J. | title = Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | chapter = 6. Gamma function and related functions | editor1-last = Abramowitz | editor1-first = Milton | editor1-link = Milton Abramowitz | editor2-last = Stegun | editor2-first = Irene A. | editor2-link = Irene Stegun | publisher = [[Dover Publications]] | location = New York | isbn = 978-0-486-61272-0 | year = 1972 | url = https://archive.org/details/handbookofmathe000abra/page/258/mode/2up?view=theater | page = 258 }}. Specifically, see 6.2 Beta Function.</ref> A key property of the beta function is its close relationship to the [[gamma function]]:<ref name=Davis622/> :<math> \Beta(z_1,z_2)=\frac{\Gamma(z_1)\,\Gamma(z_2)}{\Gamma(z_1+z_2)}</math> A proof is given below in {{slink||Relationship to the gamma function}}. The beta function is also closely related to [[binomial coefficient]]s. When {{mvar|m}} (or {{mvar|n}}, by symmetry) is a positive integer, it follows from the definition of the gamma function {{math|Ξ}} that<ref name=Davis622/> :<math> \Beta(m,n) =\frac{(m-1)!\,(n-1)!}{(m+n-1)!} = \frac{m + n}{mn} \Bigg/ \binom{m + n}{m} </math>
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