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 InverseJacobiSD

 http://functions.wolfram.com/09.47.20.0006.02

 Input Form

 D[InverseJacobiSD[z, m], {m, 2}] == (1/(4 (-1 + m)^2 m^2)) ((-2 + 4 m) EllipticE[JacobiAmplitude[InverseJacobiSD[z, m], m], m] + (-1 + m) EllipticF[JacobiAmplitude[InverseJacobiSD[z, m], m], m] + 3 (-1 + m)^2 InverseJacobiSD[z, m] + (1/(1 + (-1 + m) z^2)^2) (m JacobiCN[InverseJacobiSD[z, m], m] (-((1/(1 + m z^2)) (z^2 (-1 + z^2 + m (3 + (-7 + 8 m) z^2 + (-1 + m) (-2 + 5 m) z^4)) JacobiDS[InverseJacobiSD[z, m], m])) + z^2 (-1 + m + (-1 + m)^2 z^2) Sqrt[1/(1 + m z^2)] JacobiSN[InverseJacobiSD[z, m], m])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "2"]], "}"]]], RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", " ", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"], " ", SuperscriptBox["m", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["4", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"], " ", RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], ")"]], "2"]], RowBox[List["(", RowBox[List["m", " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"], "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["8", " ", "m"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["5", " ", "m"]]]], ")"]], " ", SuperscriptBox["z", "4"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], ")"]]]], " ", ")"]]]]]], ")"]]]]]]]]

 MathML Form

 2 sd - 1 ( z m ) m 2 1 4 ( m - 1 ) 2 m 2 ( 3 sd - 1 ( z m ) ( m - 1 ) 2 + F ( am ( sd - 1 ( z m ) m ) m ) ( m - 1 ) + ( 4 m - 2 ) E ( am ( sd - 1 ( z m ) m ) m ) + 1 ( ( m - 1 ) z 2 + 1 ) 2 ( m cn ( sd - 1 ( z m ) m ) ( z 2 ( ( m - 1 ) 2 z 2 + m - 1 ) 1 m z 2 + 1 sn ( sd - 1 ( z m ) m ) - z 2 ( z 2 + m ( ( m - 1 ) ( 5 m - 2 ) z 4 + ( 8 m - 7 ) z 2 + 3 ) - 1 ) ds ( sd - 1 ( z m ) m ) m z 2 + 1 ) ) ) m 2 InverseJacobiSD z m 1 4 m -1 2 m 2 -1 3 InverseJacobiSD z m m -1 2 EllipticF JacobiAmplitude InverseJacobiSD z m m m m -1 4 m -2 EllipticE JacobiAmplitude InverseJacobiSD z m m m 1 m -1 z 2 1 2 -1 m JacobiCN InverseJacobiSD z m m z 2 m -1 2 z 2 m -1 1 m z 2 1 -1 1 2 JacobiSN InverseJacobiSD z m m -1 z 2 z 2 m m -1 5 m -2 z 4 8 m -7 z 2 3 -1 JacobiDS InverseJacobiSD z m m m z 2 1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["m_", ",", "2"]], "}"]]]]], RowBox[List["InverseJacobiSD", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["4", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"], " ", RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List["m", " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"], "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["8", " ", "m"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["5", " ", "m"]]]], ")"]], " ", SuperscriptBox["z", "4"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], ")"]], "2"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"], " ", SuperscriptBox["m", "2"]]]]]]]]

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

 2001-10-29

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