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variants of this functions
BetaRegularized




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.22.20.0006.01









  


  








Input Form





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








Standard Form





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MathML Form







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Rule Form





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





2001-10-29