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variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z1,z2,a,b] > Differentiation > Low-order differentiation > With respect to b





http://functions.wolfram.com/06.20.20.0008.01









  


  








Input Form





D[Beta[Subscript[z, 1], Subscript[z, 2], a, b], {b, 2}] == Beta[1 - Subscript[z, 1], b, a] Log[1 - Subscript[z, 1]]^2 - Beta[1 - Subscript[z, 2], b, a] Log[1 - Subscript[z, 2]]^2 + 2 (1 - Subscript[z, 1])^b Gamma[b]^2 (Gamma[b] HypergeometricPFQRegularized[{b, b, b, 1 - a}, {1 + b, 1 + b, 1 + b}, 1 - Subscript[z, 1]] - HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - Subscript[z, 1]] Log[1 - Subscript[z, 1]]) - 2 (1 - Subscript[z, 2])^b Gamma[b]^2 (Gamma[b] HypergeometricPFQRegularized[{b, b, b, 1 - a}, {1 + b, 1 + b, 1 + b}, 1 - Subscript[z, 2]] - HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - Subscript[z, 2]] Log[1 - Subscript[z, 2]])








Standard Form





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MathML Form







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<times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> 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<power /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#915; </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29