html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 InverseBetaRegularized

 http://functions.wolfram.com/06.23.20.0006.01

 Input Form

 D[InverseBetaRegularized[z, a, b], {b, 2}] == w^(1 - 2 a) (-2 (-1 + w) w^a Gamma[b]^3 HypergeometricPFQRegularized[ {b, b, b, 1 - a}, {1 + b, 1 + b, 1 + b}, 1 - w] + (1 - w)^(1 - b) w^a Beta[1 - w, b, a] (PolyGamma[b] - PolyGamma[a + b]) (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) + (1 - a) (1 - w)^(2 - b) Gamma[b]^2 HypergeometricPFQRegularized[ {b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] ((1 - w)^b Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] - Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])) - (1 - w)^(2 - b) w Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] ((1 - w)^b Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] - Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])) + (-1 + a) (1 - w)^(2 - 2 b) Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) ((1 - w)^b Gamma[b]^2 HypergeometricPFQRegularized[ {b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] - Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])) + ((-1 + w) Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) ((-1 + b) (1 - w)^b w Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] + (1 - w)^b w^a Log[1 - w] - (-1 + b) w Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])))/(1 - w)^(2 b) + ((-1 + w) Beta[1 - w, b, a] ((-(1 - w)^b) w Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] + w Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) + (1 - w)^b w^a (PolyGamma[1, b] - PolyGamma[1, a + b])))/ (1 - w)^(2 b)) /; w == InverseBetaRegularized[z, a, b]

 Standard Form

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 MathML Form

 Rule Form

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

 2001-10-29