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variants of this functions
Hypergeometric2F1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Specific values > For rational parameters with denominators 4 and fixed z > For fixed z and a=-9/4, b>=a > For fixed z and a=-9/4, b=19/4





http://functions.wolfram.com/07.23.03.aelr.01









  


  










Input Form





Hypergeometric2F1[-(9/4), 19/4, 1/2, z] == (1/(6468 Pi^(3/2))) (((2 (1617 - 35109 z + 129664 z^2 - 161792 z^3 + 65536 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^2 + (2 (1617 - 35109 z + 129664 z^2 - 161792 z^3 + 65536 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^2 + (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z]))) ((-1617 + 5781 Sqrt[z] + 29328 z - 52864 z^(3/2) - 76800 z^2 + 112640 z^(5/2) + 49152 z^3 - 65536 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z]) (1 + Sqrt[z])^2)) ((1617 + 5781 Sqrt[z] - 29328 z - 52864 z^(3/2) + 76800 z^2 + 112640 z^(5/2) - 49152 z^3 - 65536 z^(7/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)










Standard Form





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MathML Form







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<cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["9", "4"]]], ",", FractionBox["19", "4"], ",", FractionBox["1", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1617", "-", RowBox[List["35109", " ", "z"]], "+", RowBox[List["129664", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["161792", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["65536", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1617", "-", RowBox[List["35109", " ", "z"]], "+", RowBox[List["129664", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["161792", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["65536", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1617"]], "+", RowBox[List["5781", " ", SqrtBox["z"]]], "+", RowBox[List["29328", " ", "z"]], "-", RowBox[List["52864", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["76800", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["112640", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["49152", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["65536", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1617", "+", RowBox[List["5781", " ", SqrtBox["z"]]], "-", RowBox[List["29328", " ", "z"]], "-", RowBox[List["52864", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["76800", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["112640", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["49152", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["65536", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "2"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], "2"]]], RowBox[List["6468", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]










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





2007-05-02