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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0062.01

 Input Form

 Hypergeometric2F1Regularized[a, b, c, z] \[Proportional] Piecewise[{{0, Element[-c, Integers] && -c >= 0 && ((Element[-a, Integers] && -a >= 0 && c - a <= 0) && (Element[-b, Integers] && -b >= 0 && c - b <= 0))}, {((a + b - c - 1)!/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b) + ((-1)^(a + b - c - 1)/((a + b - c)! Gamma[c - a] Gamma[c - b])) (Log[1 - z] + EulerGamma + PolyGamma[a] + PolyGamma[b] - PolyGamma[1 + a + b - c]), Element[a + b - c, Integers] && a + b - c > 0}, {(-(1/(Gamma[a] Gamma[b]))) (Log[1 - z] + 2 EulerGamma + PolyGamma[a] + PolyGamma[b]), c == a + b}, {(c - a - b - 1)!/(Gamma[c - a] Gamma[c - b]) - (1/((c - a - b)! Gamma[a] Gamma[b])) (z - 1)^(c - a - b) (Log[1 - z] + EulerGamma - PolyGamma[1 + c - a - b] + PolyGamma[c - a] + PolyGamma[c - b]), Element[c - a - b, Integers] && c - a - b > 0}}, Gamma[c - a - b]/(Gamma[c - a] Gamma[c - b]) + (Gamma[a + b - c]/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b)] /; (z -> 1)

 Standard Form

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 MathML Form

 2 F ~ 1 ( a , b ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1Regularized] 0 - c ( ( - a c - a 0 ) ( - c - b 0 ) ) ( a + b - c - 1 ) ! ( 1 - z ) c - a - b Γ ( a ) Γ ( b ) + ( - 1 ) a + b - c - 1 ( log ( 1 - z ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) + ψ TagBox["\[Psi]", PolyGamma] ( b ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b - c + 1 ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( a + b - c ) ! Γ ( c - a ) Γ ( c - b ) a + b - c + - log ( 1 - z ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) + ψ TagBox["\[Psi]", PolyGamma] ( b ) + 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] Γ ( a ) Γ ( b ) c a + b ( c - a - b - 1 ) ! Γ ( c - a ) Γ ( c - b ) - ( log ( 1 - z ) + ψ TagBox["\[Psi]", PolyGamma] ( c - a ) - ψ TagBox["\[Psi]", PolyGamma] ( c - a - b + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( c - b ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( z - 1 ) c - a - b ( c - a - b ) ! Γ ( a ) Γ ( b ) c - a - b + Γ ( a + b - c ) ( 1 - z ) c - a - b Γ ( a ) Γ ( b ) + Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( z "\[Rule]" 1 ) Condition Proportional Hypergeometric2F1Regularized a b c z 0 -1 c -1 a c -1 a 0 -1 c -1 b 0 a b -1 c -1 1 -1 z c -1 a -1 b Gamma a Gamma b -1 -1 a b -1 c -1 1 -1 z PolyGamma a PolyGamma b -1 PolyGamma a b -1 c 1 a b -1 c Gamma c -1 a Gamma c -1 b -1 a b -1 c SuperPlus -1 1 -1 z PolyGamma a PolyGamma b 2 Gamma a Gamma b -1 c a b c -1 a -1 b -1 Gamma c -1 a Gamma c -1 b -1 -1 1 -1 z PolyGamma c -1 a -1 PolyGamma c -1 a -1 b 1 PolyGamma c -1 b z -1 c -1 a -1 b c -1 a -1 b Gamma a Gamma b -1 c -1 a -1 b SuperPlus Gamma a b -1 c 1 -1 z c -1 a -1 b Gamma a Gamma b -1 Gamma c -1 a -1 b Gamma c -1 a Gamma c -1 b -1 Rule z 1 [/itex]

 Rule Form

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

 2007-05-02