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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0063.01

 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))}, {Gamma[c - a - b]/(Gamma[c - a] Gamma[c - b]), Re[c - a - b] > 0}, {(Gamma[a + b - c]/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b), Re[c - a - b] < 0}, {-(Log[1 - z]/(Gamma[a] Gamma[b])), c == a + b}}, 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 ) ( - b c - b 0 ) ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) Re ( c - a - b ) > 0 Γ ( a + b - c ) ( 1 - z ) c - a - b Γ ( a ) Γ ( b ) Re ( c - a - b ) < 0 - log ( 1 - z ) 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 b c -1 b 0 Gamma c -1 a -1 b Gamma c -1 a Gamma c -1 b -1 c -1 a -1 b 0 Gamma a b -1 c 1 -1 z c -1 a -1 b Gamma a Gamma b -1 c -1 a -1 b 0 -1 1 -1 z Gamma a Gamma b -1 c a b 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