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; }

 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.26.0209.01

 Input Form

 (Sqrt[z] + Sqrt[1 + z])^((1/2) (-1 + a + b)) Hypergeometric2F1Regularized[a, b, (1/2) (1 + a + b), (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] Hypergeometric2F1[(1/2) (1 - a + b), (1/2) (1 + a - b), (1/2) (3 - a - b), (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] == (1/Pi^(3/2)) 2^((1/2) (-1 + a + b)) Cos[(1/2) (a - b) Pi] Gamma[(1/2) (3 - a - b)] MeijerG[{{(1/4) (3 + a + b)}, {(1/4) (1 + 3 a + 3 b), (1/4) (5 - a - b)}}, {{(1/4) (1 + a + b), (1/4) (1 + 3 a - b), (1/4) (1 - a + 3 b)}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b"]], ")"]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b"]], ")"]]]], ",", FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], RowBox[List["2", SqrtBox["z"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "a", "+", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "-", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "a", "-", "b"]], ")"]]]], ",", FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], RowBox[List["2", SqrtBox["z"]]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]], SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b"]], ")"]]]]], RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "a", "-", "b"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", "a", "+", "b"]], ")"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "a"]], "+", RowBox[List["3", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", "a", "-", "b"]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "a"]], "-", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", "a", "+", RowBox[List["3", " ", "b"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]

 MathML Form

 ( z + z + 1 ) a + b - 1 2 2 F ~ 1 ( a , b ; a + b + 1 2 ; z - z + 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[FractionBox[RowBox[List["a", "+", "b", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["z", "+", "1"]]]]], RowBox[List["2", " ", SqrtBox["z"]]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] 2 F 1 ( b - a + 1 2 , a - b + 1 2 ; 3 - a - b 2 ; z - z + 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[FractionBox[RowBox[List["b", "-", "a", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "-", "b", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["3", "-", "a", "-", "b"]], "2"], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["z", "+", "1"]]]]], RowBox[List["2", " ", SqrtBox["z"]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] 2 a + b - 1 2 π 3 / 2 cos ( 1 2 ( a - b ) π ) Γ ( 1 2 ( 3 - a - b ) ) G 3 , 3 3 , 1 ( z a + b + 3 4 , 3 a + 3 b + 1 4 , 5 - a - b 4 a + b + 1 4 , 3 a - b + 1 4 , 3 b - a + 1 4 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["3", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["a", "+", "b", "+", "3"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "a"]], "+", RowBox[List["3", "b"]], "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["5", "-", "a", "-", "b"]], "4"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List["a", "+", "b", "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "a"]], "-", "b", "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "b"]], "-", "a", "+", "1"]], "4"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] /; z ( - 1 , 0 ) ( z + z + 1 ) a + b - 1 2 2 F ~ 1 ( a , b ; a + b + 1 2 ; z - z + 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[FractionBox[RowBox[List["a", "+", "b", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["z", "+", "1"]]]]], RowBox[List["2", " ", SqrtBox["z"]]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] 2 F 1 ( b - a + 1 2 , a - b + 1 2 ; 3 - a - b 2 ; z - z + 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[FractionBox[RowBox[List["b", "-", "a", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "-", "b", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["3", "-", "a", "-", "b"]], "2"], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["z", "+", "1"]]]]], RowBox[List["2", " ", SqrtBox["z"]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] 2 a + b - 1 2 π 3 / 2 cos ( 1 2 ( a - b ) π ) Γ ( 1 2 ( 3 - a - b ) ) G 3 , 3 3 , 1 ( z a + b + 3 4 , 3 a + 3 b + 1 4 , 5 - a - b 4 a + b + 1 4 , 3 a - b + 1 4 , 3 b - a + 1 4 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["3", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["a", "+", "b", "+", "3"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "a"]], "+", RowBox[List["3", "b"]], "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["5", "-", "a", "-", "b"]], "4"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List["a", "+", "b", "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "a"]], "-", "b", "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "b"]], "-", "a", "+", "1"]], "4"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] /; z ( - 1 , 0 ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["z_"], "+", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a_", "+", "b_"]], ")"]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a_", "+", "b_"]], ")"]]]], ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "a_", "+", "b_"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a_", "-", "b_"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "a_", "-", "b_"]], ")"]]]], ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b"]], ")"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "a", "-", "b"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", "a", "+", "b"]], ")"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "a"]], "+", RowBox[List["3", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", "a", "-", "b"]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "a"]], "-", "b"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", "a", "+", RowBox[List["3", " ", "b"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]

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

 2001-10-29