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 InverseJacobiNS

 http://functions.wolfram.com/09.45.20.0007.02

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], " ", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]], RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "m"]], ")"]], RowBox[List["1", "+", "j", "-", "n"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "n"]], "]"]]]], ")"]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "j"]], ",", RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "j", "-", "n"]], "2"], ",", FractionBox[RowBox[List["1", "+", "j", "-", "n"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "n"]], ",", RowBox[List["1", "-", FractionBox["m", SuperscriptBox["z", "2"]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n ns - 1 ( z m ) z n δ KroneckerDelta n ns - 1 ( z m ) - 2 n - 1 π z n - 1 ( n - 1 ) ! cd ( ns - 1 ( z m ) m ) z 2 - 1 j = 0 n - 1 ( z 2 - 1 ) - j ( 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 ; 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["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["m", SuperscriptBox["z", "2"]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; n Condition z n InverseJacobiNS z m KroneckerDelta n InverseJacobiNS z m -1 2 n -1 z n -1 n -1 JacobiCD InverseJacobiNS z m m z 2 -1 -1 j 0 n -1 z 2 -1 -1 j z 2 -1 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 1 -1 m z 2 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["InverseJacobiNS", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], " ", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]], RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "m"]], ")"]], RowBox[List["1", "+", "j", "-", "n"]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "j"]], ",", RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "j", "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "j", "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "n"]], ",", RowBox[List["1", "-", FractionBox["m", SuperscriptBox["z", "2"]]]]]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "n"]], "]"]]]]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29