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 InverseJacobiNC

 http://functions.wolfram.com/09.43.20.0007.01

 Input Form

 D[InverseJacobiNC[z, m], {z, n}] == ((2^(-1 + n) Pi z^(-1 + n) (-1 + n)! JacobiDS[InverseJacobiNC[z, m], m])/ (m + z^2 - m z^2)) Sum[(((1 - m)^(n - j - 1) (m + (1 - m) z^2)^(1 + j - n))/ ((z^2 - 1)^j (j! (n - j - 1)! Gamma[1/2 - j] Gamma[3/2 + j - n]))) Hypergeometric2F1[(1 - j)/2, -(j/2), 1/2 - j, 1 - 1/z^2] Hypergeometric2F1[(2 + j - n)/2, (1 + j - n)/2, 3/2 + j - n, 1 + m/((1 - m) z^2)], {j, 0, n - 1}] /; Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 n nc - 1 ( z m ) z n 2 n - 1 π z n - 1 ( n - 1 ) ! ds ( nc - 1 ( z m ) m ) - m z 2 + z 2 + m j = 0 n - 1 ( 1 - m ) n - j - 1 ( z 2 - 1 ) - j ( ( 1 - m ) z 2 + m ) j - n + 1 j ! ( n - j - 1 ) ! Γ ( 1 2 - j ) Γ ( j - n + 3 2 ) 2 F 1 ( 1 - j 2 , - j 2 ; 1 2 - j ; 1 - 1 z 2 ) 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[FractionBox[RowBox[List["1", "-", "j"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["j", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "j"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] 2 F 1 ( j - n + 2 2 , j - n + 1 2 ; j - n + 3 2 ; m ( 1 - m ) z 2 + 1 ) 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[FractionBox[RowBox[List["j", "-", "n", "+", "2"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["j", "-", "n", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["j", "-", "n", "+", FractionBox["3", "2"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["m", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; n + Condition z n InverseJacobiNC z m 2 n -1 z n -1 n -1 JacobiDS InverseJacobiNC z m m -1 m z 2 z 2 m -1 j 0 n -1 1 -1 m n -1 j -1 z 2 -1 -1 j 1 -1 m z 2 m j -1 n 1 j n -1 j -1 Gamma 1 2 -1 j Gamma j -1 n 3 2 -1 Hypergeometric2F1 1 -1 j 2 -1 -1 j 2 -1 1 2 -1 j 1 -1 1 z 2 -1 Hypergeometric2F1 j -1 n 2 2 -1 j -1 n 1 2 -1 j -1 n 3 2 m 1 -1 m z 2 -1 1 n SuperPlus [/itex]

 Rule Form

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

 2001-10-29