Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9736)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-23/4, b>=a

For fixed z and a=-23/4, b=9/4

http://functions.wolfram.com/07.23.03.9736.01

Input Form

Hypergeometric2F1[-(23/4), 9/4, 6, z] == (1/(551032827345 Pi z^5)) (16384 (-2 Sqrt[1 - z] (-894976 + 6111008 z - 15474607 z^2 + 12517428 z^3 + 15545838 z^4 - 81910108 z^5 + 126922761 z^6 - 107936064 z^7 + 54212096 z^8 - 15171584 z^9 + 1835008 z^10) EllipticE[(1/2) (1 - Sqrt[1 - z])] + (-894976 + 6782240 z - 19994935 z^2 + 23722545 z^3 + 7066290 z^4 - 26793046 z^5 + 37843365 z^6 - 30342555 z^7 + 14560640 z^8 - 3921920 z^9 + 458752 z^10) EllipticK[(1/2) (1 - Sqrt[1 - z])] + Sqrt[1 - z] (-894976 + 6111008 z - 15474607 z^2 + 12517428 z^3 + 15545838 z^4 - 81910108 z^5 + 126922761 z^6 - 107936064 z^7 + 54212096 z^8 - 15171584 z^9 + 1835008 z^10) EllipticK[(1/2) (1 - Sqrt[1 - z])]))

Standard Form

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MathML Form

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type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </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["23", "4"]]], ",", FractionBox["9", "4"], ",", "6", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["16384", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "894976"]], "+", RowBox[List["6111008", " ", "z"]], "-", RowBox[List["15474607", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["12517428", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["15545838", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["81910108", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["126922761", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["107936064", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["54212096", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["15171584", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["1835008", " ", SuperscriptBox["z", "10"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "894976"]], "+", RowBox[List["6782240", " ", "z"]], "-", RowBox[List["19994935", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["23722545", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["7066290", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["26793046", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["37843365", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["30342555", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["14560640", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["3921920", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["458752", " ", SuperscriptBox["z", "10"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "894976"]], "+", RowBox[List["6111008", " ", "z"]], "-", RowBox[List["15474607", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["12517428", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["15545838", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["81910108", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["126922761", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["107936064", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["54212096", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["15171584", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["1835008", " ", SuperscriptBox["z", "10"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["551032827345", " ", "\[Pi]", " ", SuperscriptBox["z", "5"]]]]]]]]

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

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]