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 InverseJacobiNS

 http://functions.wolfram.com/09.45.20.0013.01

 Input Form

 D[InverseJacobiNS[z, m], {m, 3}] == (1/(8 (-1 + m)^3 m^3)) ((-8 - 23 (-1 + m) m) EllipticE[JacobiAmplitude[InverseJacobiNS[z, m], m], m] - (-1 + m) (-7 + 11 m) EllipticF[JacobiAmplitude[ InverseJacobiNS[z, m], m], m] + (1/(m - z^2)^3) (-15 (-1 + m)^3 (m - z^2)^3 InverseJacobiNS[z, m] + m z ((1/z^2) ((m - z^2) (m^2 (9 + m (-24 + 23 m)) - m (11 + 5 m (-6 + 7 m)) z^2 + (5 + 3 m (-4 + 5 m)) z^4) JacobiCD[InverseJacobiNS[z, m], m]) - (-1 + m) Sqrt[1 - m/z^2] (m - z^2)^2 JacobiCN[InverseJacobiNS[z, m], m])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "3"]], "}"]]], RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["m", "3"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "8"]], "-", RowBox[List["23", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["11", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], "3"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "15"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], "3"], " ", RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List["m", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["z", "2"]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "24"]], "+", RowBox[List["23", " ", "m"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["m", " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["5", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["7", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["5", "+", RowBox[List["3", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", RowBox[List["5", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["m", SuperscriptBox["z", "2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], "2"], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 3 ns - 1 ( z m ) m 3 1 8 ( m - 1 ) 3 m 3 ( ( - 23 ( m - 1 ) m - 8 ) E ( am ( ns - 1 ( z m ) m ) m ) - ( m - 1 ) ( 11 m - 7 ) F ( am ( ns - 1 ( z m ) m ) m ) + 1 ( m - z 2 ) 3 ( m z ( 1 z 2 ( m - z 2 ) ( ( 3 m ( 5 m - 4 ) + 5 ) z 4 - m ( 5 m ( 7 m - 6 ) + 11 ) z 2 + m 2 ( m ( 23 m - 24 ) + 9 ) ) cd ( ns - 1 ( z m ) m ) - ( m - 1 ) 1 - m z 2 ( m - z 2 ) 2 cn ( ns - 1 ( z m ) m ) ) - 15 ( m - 1 ) 3 ( m - z 2 ) 3 ns - 1 ( z m ) ) ) m 3 InverseJacobiNS z m 1 8 m -1 3 m 3 -1 -23 m -1 m -8 EllipticE JacobiAmplitude InverseJacobiNS z m m m -1 m -1 11 m -7 EllipticF JacobiAmplitude InverseJacobiNS z m m m 1 m -1 z 2 3 -1 m z 1 z 2 -1 m -1 z 2 3 m 5 m -4 5 z 4 -1 m 5 m 7 m -6 11 z 2 m 2 m 23 m -24 9 JacobiCD InverseJacobiNS z m m -1 m -1 1 -1 m z 2 -1 1 2 m -1 z 2 2 JacobiCN InverseJacobiNS z m m -1 15 m -1 3 m -1 z 2 3 InverseJacobiNS z m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["m_", ",", "3"]], "}"]]]]], RowBox[List["InverseJacobiNS", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "8"]], "-", RowBox[List["23", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["11", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "15"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], "3"], " ", RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List["m", " ", "z", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "24"]], "+", RowBox[List["23", " ", "m"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["m", " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["5", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["7", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["5", "+", RowBox[List["3", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", RowBox[List["5", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], SuperscriptBox["z", "2"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["m", SuperscriptBox["z", "2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], "2"], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], ")"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], "3"]]]], RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["m", "3"]]]]]]]]

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

 2007-05-02