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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.26.0259.01

 Input Form

 (z + Sqrt[1 + z^2])^(a + b - c) Hypergeometric2F1Regularized[a, b, c, (z - Sqrt[1 + z^2])/(2 z)] Hypergeometric2F1[-a + c, -b + c, 1 - a - b + c, (z - Sqrt[1 + z^2])/(2 z)] == ((2^(-1 + c) Gamma[1 - a - b + c])/(Sqrt[Pi] Gamma[-a + c] Gamma[-b + c])) MeijerG[{{1 + (a + b - c)/2}, {(3 c - a - b)/2, (a + b + c)/2, (c - a - b)/2 + 1}}, {{(-a + b + c)/2, (a - b + c)/2, c/2, (1 + c)/2}, {}}, z, 1/2] /; Re[z] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["a", "+", "b", "-", "c"]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", "z"]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ",", RowBox[List["1", "-", "a", "-", "b", "+", "c"]], ",", FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", "z"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "c"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "-", "b", "+", "c"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["a", "+", "b", "-", "c"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", "c"]], "-", "a", "-", "b"]], "2"], ",", FractionBox[RowBox[List["a", "+", "b", "+", "c"]], "2"], ",", RowBox[List[FractionBox[RowBox[List["c", "-", "a", "-", "b"]], "2"], "+", "1"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", "c"]], "2"], ",", FractionBox[RowBox[List["a", "-", "b", "+", "c"]], "2"], ",", FractionBox["c", "2"], " ", ",", FractionBox[RowBox[List["1", "+", "c"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]

 MathML Form

 ( z + z 2 + 1 ) a + b - c 2 F ~ 1 ( a , b ; c ; z - z 2 + 1 2 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]], RowBox[List["2", " ", "z"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] 2 F 1 ( c - a , c - b ; c - a - b + 1 ; z - z 2 + 1 2 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["c", "-", "a"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "-", "b"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", "a", "-", "b", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]], RowBox[List["2", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] 2 c - 1 Γ ( c - a - b + 1 ) π Γ ( c - a ) Γ ( c - b ) G 4 , 4 4 , 1 ( z , 1 2 a + b - c 2 + 1 , 3 c - a - b 2 , a + b + c 2 , c - a - b 2 + 1 b - a + c 2 , a - b + c 2 , c 2 , c + 1 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["4", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["a", "+", "b", "-", "c"]], "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "c"]], "-", "a", "-", "b"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "b", "+", "c"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", "a", "-", "b"]], "2"], "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List["b", "-", "a", "+", "c"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "-", "b", "+", "c"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["c", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["c", "+", "1"]], "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] /; Re ( z ) > 0 Condition z z 2 1 1 2 a b -1 c Hypergeometric2F1Regularized a b c z -1 z 2 1 1 2 2 z -1 Hypergeometric2F1 c -1 a c -1 b c -1 a -1 b 1 z -1 z 2 1 1 2 2 z -1 2 c -1 Gamma c -1 a -1 b 1 1 2 Gamma c -1 a Gamma c -1 b -1 MeijerG a b -1 c 2 -1 1 3 c -1 a -1 b 2 -1 a b c 2 -1 c -1 a -1 b 2 -1 1 b -1 a c 2 -1 a -1 b c 2 -1 c 2 -1 c 1 2 -1 z 1 2 z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z_", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], ")"]], RowBox[List["a_", "+", "b_", "-", "c_"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", FractionBox[RowBox[List["z_", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], RowBox[List["2", " ", "z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "a_"]], "+", "c_"]], ",", RowBox[List[RowBox[List["-", "b_"]], "+", "c_"]], ",", RowBox[List["1", "-", "a_", "-", "b_", "+", "c_"]], ",", FractionBox[RowBox[List["z_", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], RowBox[List["2", " ", "z_"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "c"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "-", "b", "+", "c"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", "c"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "c"]], "-", "a", "-", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", "c"]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["c", "-", "a", "-", "b"]], ")"]]]], "+", "1"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", "c"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", "c"]], ")"]]]], ",", FractionBox["c", "2"], ",", FractionBox[RowBox[List["1", "+", "c"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], "]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]

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

 2001-10-29