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; }

 Hypergeometric2F1

 http://functions.wolfram.com/07.23.26.0206.01

 Input Form

 (Sqrt[z] + Sqrt[1 + z])^(1/2 + 2 a - c) Hypergeometric2F1[a, 1/2 + a, c, (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] Hypergeometric2F1Regularized[ -(1/2) - a + c, -a + c, 1/2 - 2 a + c, (Sqrt[z] - Sqrt[1 + z])/ (2 Sqrt[z])] == ((2^(-3 - 2 a + 3 c) Gamma[c])/ (Pi Gamma[-1 - 2 a + 2 c])) MeijerG[{{(5 - 2 c)/4 + a}, {(1 + 2 c)/4 + a, (6 c - 1)/4 - a, (3 + 2 c)/4 - a}}, {{(1/4) (-1 + 2 c), c/2, (1/4) (1 + 2 c), (1 + c)/2}, {}}, 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["2", " ", "a"]], "-", "c"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List[FractionBox["1", "2"], "+", "a"]], ",", "c", ",", FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], RowBox[List["2", SqrtBox["z"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "a", "+", "c"]], ",", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "a"]], "+", "c"]], ",", FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], RowBox[List["2", SqrtBox["z"]]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["3", " ", "c"]]]]], " ", RowBox[List["Gamma", "[", "c", "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "c"]]]], "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["5", "-", RowBox[List["2", "c"]]]], "4"], "+", "a"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["2", "c"]]]], "4"], "+", "a"]], ",", RowBox[List[FractionBox[RowBox[List[RowBox[List["6", "c"]], "-", "1"]], "4"], "-", "a"]], ",", RowBox[List[FractionBox[RowBox[List["3", "+", RowBox[List["2", "c"]]]], "4"], " ", "-", "a"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "c"]]]], ")"]]]], ",", FractionBox["c", "2"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "c"]]]], ")"]]]], ",", FractionBox[RowBox[List["1", "+", "c"]], "2"]]], "}"]], ",", 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 ) 2 a - c + 1 2 2 F 1 ( a , a + 1 2 ; c ; 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["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", 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 F ~ 1 ( c - a - 1 2 , c - a ; c - 2 a + 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[RowBox[List["c", "-", "a", "-", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "-", "a"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", RowBox[List["2", "a"]], "+", FractionBox["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 3 c - 2 a - 3 Γ ( c ) π Γ ( 2 c - 2 a - 1 ) G 4 , 4 4 , 1 ( z a + 5 - 2 c 4 , a + 2 c + 1 4 , 6 c - 1 4 - a , 2 c + 3 4 - a 2 c - 1 4 , c 2 , 2 c + 1 4 , c + 1 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["4", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["a", "+", FractionBox[RowBox[List["5", "-", RowBox[List["2", "c"]]]], "4"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox[RowBox[List[RowBox[List["2", "c"]], "+", "1"]], "4"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["6", "c"]], "-", "1"]], "4"], "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "c"]], "+", "3"]], "4"], "-", "a"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["2", "c"]], "-", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["c", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["2", "c"]], "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["c", "+", "1"]], "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] /; z ( - 1 , 0 ) ( z + z + 1 ) 2 a - c + 1 2 2 F 1 ( a , a + 1 2 ; c ; 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["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", 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 F ~ 1 ( c - a - 1 2 , c - a ; c - 2 a + 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[RowBox[List["c", "-", "a", "-", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "-", "a"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", RowBox[List["2", "a"]], "+", FractionBox["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 3 c - 2 a - 3 Γ ( c ) π Γ ( 2 c - 2 a - 1 ) G 4 , 4 4 , 1 ( z a + 5 - 2 c 4 , a + 2 c + 1 4 , 6 c - 1 4 - a , 2 c + 3 4 - a 2 c - 1 4 , c 2 , 2 c + 1 4 , c + 1 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["4", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["a", "+", FractionBox[RowBox[List["5", "-", RowBox[List["2", "c"]]]], "4"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox[RowBox[List[RowBox[List["2", "c"]], "+", "1"]], "4"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["6", "c"]], "-", "1"]], "4"], "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "c"]], "+", "3"]], "4"], "-", "a"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["2", "c"]], "-", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["c", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["2", "c"]], "+", "1"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["c", "+", "1"]], "2"], 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["2", " ", "a_"]], "-", "c_"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", RowBox[List[FractionBox["1", "2"], "+", "a_"]], ",", "c_", ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "a_", "+", "c_"]], ",", RowBox[List[RowBox[List["-", "a_"]], "+", "c_"]], ",", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "a_"]], "+", "c_"]], ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["3", " ", "c"]]]]], " ", RowBox[List["Gamma", "[", "c", "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "c"]]]], ")"]]]], "+", "a"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "c"]]]], ")"]]]], "+", "a"]], ",", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", "c"]], "-", "1"]], ")"]]]], "-", "a"]], ",", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "c"]]]], ")"]]]], "-", "a"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "c"]]]], ")"]]]], ",", FractionBox["c", "2"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "c"]]]], ")"]]]], ",", FractionBox[RowBox[List["1", "+", "c"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "c"]]]], "]"]]]]], "/;", 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