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 InverseJacobiSD

 http://functions.wolfram.com/09.47.20.0007.01

 Input Form

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

 Standard Form

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 MathML Form

 n sd - 1 ( z m ) z n 2 n - 1 π z n - 1 ( n - 1 ) ! cn ( sd - 1 ( z m ) m ) ( m - 1 ) z 2 + 1 j = 0 n - 1 ( m - 1 ) n - j - 1 m j ( ( m - 1 ) z 2 + 1 ) j - n + 1 ( m z 2 + 1 ) - j 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 m 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", RowBox[List["m", " ", 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 ; 1 + 1 ( m - 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["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["1", "+", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; n + Condition z n InverseJacobiSD z m 2 n -1 z n -1 n -1 JacobiCN InverseJacobiSD z m m m -1 z 2 1 -1 j 0 n -1 m -1 n -1 j -1 m j m -1 z 2 1 j -1 n 1 m z 2 1 -1 j 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 m z 2 -1 Hypergeometric2F1 j -1 n 2 2 -1 j -1 n 1 2 -1 j -1 n 3 2 1 1 m -1 z 2 -1 n SuperPlus [/itex]

 Rule Form

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

 2001-10-29