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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0066.01

 Input Form

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

 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] ( b - a - 1 ) ! ( - z ) - a Γ ( b ) Γ ( c - a ) + 1 π ( b - a ) ! Γ ( a ) sin ( ( c - a ) π ) Γ ( b - c + 1 ) ( log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( a + b - a + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( c - b ) - ψ TagBox["\[Psi]", PolyGamma] ( b ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( - z ) - b b - a + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] c - a TagBox[RowBox[List["\[NotElement]", TagBox["\[DoubleStruckCapitalZ]", Function[Integers]]]], Function[List[], Integers]] ( - z ) - a ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( c - a ) - ψ TagBox["\[Psi]", PolyGamma] ( a ) - 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) Γ ( a ) Γ ( c - a ) b a c - a TagBox[RowBox[List["\[NotElement]", TagBox["\[DoubleStruckCapitalZ]", Function[Integers]]]], Function[List[], Integers]] ( a - b - 1 ) ! ( - z ) - b Γ ( a ) Γ ( c - b ) + 1 π ( a - b ) ! Γ ( b ) sin ( ( c - b ) π ) Γ ( a - c + 1 ) ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( c - a ) - ψ TagBox["\[Psi]", PolyGamma] ( a ) + ψ TagBox["\[Psi]", PolyGamma] ( a - b + 1 ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( - z ) - a a - b + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] c - b TagBox[RowBox[List["\[NotElement]", TagBox["\[DoubleStruckCapitalZ]", Function[Integers]]]], Function[List[], Integers]] ( - 1 ) a - c ( b - c ) ! ( - z ) - b Γ ( a ) ( b - a ) ! b - a TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] a - c TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] ( - 1 ) b - c ( a - c ) ! ( - z ) - a Γ ( b ) ( a - b ) ! a - b TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] b - c TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] Γ ( b - a ) ( - z ) - a Γ ( c - a ) Γ ( b ) + ( - 1 ) b - a ( log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( b - a + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( c - b ) - ψ TagBox["\[Psi]", PolyGamma] ( b ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( - z ) - b Γ ( a ) ( b - a ) ! ( c - b - 1 ) ! b - a + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] c - a + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] c - b > 0 Γ ( a - b ) ( - z ) - b Γ ( c - b ) Γ ( a ) + ( - 1 ) a - b ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( c - a ) - ψ TagBox["\[Psi]", PolyGamma] ( a ) + ψ TagBox["\[Psi]", PolyGamma] ( a - b + 1 ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( - z ) - a Γ ( b ) ( a - b ) ! ( c - a - 1 ) ! a - b + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] c - b + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] c - a > 0 ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( c - a ) - ψ TagBox["\[Psi]", PolyGamma] ( a ) - 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( - z ) - a Γ ( a ) ( c - a - 1 ) ! + Γ ( c ) ( - z ) - c Γ ( a ) 2 ( ( c - a ) ! ) 2 b a c - a + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] 0 - c ( ( - a c - a 0 ) ( - b c - b 0 ) ) ( - 1 ) a ( b ) - a TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], RowBox[List["-", "a"]]], Pochhammer] z - a ( c - a - 1 ) ! - c TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] - a TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] c - a > 0 ( - 1 ) b ( a ) - b TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["-", "b"]]], Pochhammer] z - b ( c - b - 1 ) ! - c TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] - b TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] c - b > 0 Γ ( b - a ) ( - z ) - a Γ ( b ) Γ ( c - a ) + Γ ( a - b ) ( - z ) - b Γ ( a ) Γ ( c - b ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) 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] ( b - a - 1 ) ! 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02