Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a923)

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=-19/4, b>=a

For fixed z and a=-19/4, b=-17/4

http://functions.wolfram.com/07.23.03.a923.01

Input Form

Hypergeometric2F1[-(19/4), -(17/4), 6, -z] == -((1/(3506298768697725 Pi z^5)) (16384 (1 + z)^(1/4) (2 (661504 + 14790816 z + 176715561 z^2 + 1648257695 z^3 + 17425493085 z^4 - 671902782117 z^5 + 1729718666907 z^6 - 1188973865763 z^7 + 232542132255 z^8 - 9213428535 z^9) EllipticE[1/2 - 1/(2 Sqrt[1 + z])] + (-661504 (1 + Sqrt[1 + z]) - 31008 z (477 + 461 Sqrt[1 + z]) - 4849845 z^4 (3593 + 3361 Sqrt[1 + z]) - 11305 z^3 (145799 + 134927 Sqrt[1 + z]) - 969 z^2 (182369 + 171385 Sqrt[1 + z]) + 15 z^9 (614228569 + 20156305 Sqrt[1 + z]) - 45 z^8 (5167602939 + 525833411 Sqrt[1 + z]) + 21 z^5 (31995370577 + 12187679817 Sqrt[1 + z]) - 21 z^6 (82367555567 + 21746586071 Sqrt[1 + z]) + 9 z^7 (132108207307 + 23395647155 Sqrt[1 + z])) EllipticK[1/2 - 1/(2 Sqrt[1 + z])])))

Standard Form

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MathML Form

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type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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["19", "4"]]], ",", RowBox[List["-", FractionBox["17", "4"]]], ",", "6", ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["16384", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["661504", "+", RowBox[List["14790816", " ", "z"]], "+", RowBox[List["176715561", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1648257695", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["17425493085", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["671902782117", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["1729718666907", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["1188973865763", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["232542132255", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["9213428535", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "661504"]], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", "z"]]]]], ")"]]]], "-", RowBox[List["31008", " ", "z", " ", RowBox[List["(", RowBox[List["477", "+", RowBox[List["461", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["4849845", " ", SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["3593", "+", RowBox[List["3361", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["11305", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List["145799", "+", RowBox[List["134927", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["969", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["182369", "+", RowBox[List["171385", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List["15", " ", SuperscriptBox["z", "9"], " ", RowBox[List["(", RowBox[List["614228569", "+", RowBox[List["20156305", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["45", " ", SuperscriptBox["z", "8"], " ", RowBox[List["(", RowBox[List["5167602939", "+", RowBox[List["525833411", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List["21", " ", SuperscriptBox["z", "5"], " ", RowBox[List["(", RowBox[List["31995370577", "+", RowBox[List["12187679817", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["21", " ", SuperscriptBox["z", "6"], " ", RowBox[List["(", RowBox[List["82367555567", "+", RowBox[List["21746586071", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List["9", " ", SuperscriptBox["z", "7"], " ", RowBox[List["(", RowBox[List["132108207307", "+", RowBox[List["23395647155", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["3506298768697725", " ", "\[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]