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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.0177.02

 Input Form

 Hypergeometric2F1[1, n, m + 1/2, z] == ((2 m - 1)/(2 (n - 1))) Sum[Pochhammer[1/2 - n + m, k]/ (Pochhammer[2 - n, k] (1 - z)^(k + 1)), {k, 0, n - m - 1}] + (((Gamma[1/2 + n - m] Gamma[1/2 + m])/(Pi (n - 1)!)) (1 - z)^(m - n) (((2 Sqrt[z])/Sqrt[1 - z]) ArcSin[Sqrt[z]] - Sum[((k - 1)! z^k)/Pochhammer[1/2, k], {k, 1, m - 1}]))/z^m /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0 && ((m = 1 && n > 2) || Inequality[1, Less, m, LessEqual, n])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "n", ",", RowBox[List["m", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "m"]], "-", "1"]], RowBox[List["2", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m", "-", "1"]]], FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "n", "+", "m"]], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "n"]], ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["k", "+", "1"]]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "n", "-", "m"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "m"]], "]"]]]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["m", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "m"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", SqrtBox["z"]]], SqrtBox[RowBox[List["1", "-", "z"]]]], RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "!"]], SuperscriptBox["z", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["m", "=", RowBox[List["1", "\[And]", RowBox[List["n", ">", "2"]]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List["1", "<", "m", "\[LessEqual]", "n"]], ")"]]]], ")"]]]]]]]]

 MathML Form

 2 F 1 ( 1 , n ; m + 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["1", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["n", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["m", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] Γ ( 1 2 - m + n ) Γ ( m + 1 2 ) π ( n - 1 ) ! z - m ( 1 - z ) m - n ( 2 z 1 - z sin - 1 ( z ) - k = 1 m - 1 ( k - 1 ) ! z k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ) + 2 m - 1 2 ( n - 1 ) k = 0 n - m - 1 ( m - n + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "n", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 2 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "n"]], ")"]], "k"], Pochhammer] ( 1 - z ) k + 1 /; m + n + ( ( m = 1 n > 2 ) 1 < m n ) Condition Hypergeometric2F1 1 n m 1 2 z Gamma 1 2 -1 m n Gamma m 1 2 n -1 -1 z -1 m 1 -1 z m -1 n 2 z 1 2 1 -1 z 1 2 -1 z 1 2 -1 k 1 m -1 k -1 z k Pochhammer 1 2 k -1 2 m -1 2 n -1 -1 k 0 n -1 m -1 Pochhammer m -1 n 1 2 k Pochhammer 2 -1 n k 1 -1 z k 1 -1 m SuperPlus n SuperPlus Set m 1 n 2 Inequality 1 m n [/itex]

 Rule Form

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

 2001-10-29