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 InverseJacobiSD

 http://functions.wolfram.com/09.47.07.0003.01

 Input Form

 InverseJacobiSD[z, m] == InverseJacobiSD[Subscript[z, 0], m] + ((Sqrt[1 + m z^2] JacobiCN[InverseJacobiSD[z, m], m])/ Sqrt[1 + (-1 + m) z^2]) Integrate[ 1/(Sqrt[m t^2 + 1] Sqrt[1 - (1 - m) t^2]), {t, Subscript[z, 0], z}] /; !Exists[\[Tau], {Element[\[Tau], Reals], 0 < \[Tau] < 1}, Im[m (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2 + 1] == 0 && m (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2 + 1 < 0 && Im[1 - (1 - m) (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2] == 0 && 1 - (1 - m) (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2 < 0]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List[SubscriptBox["z", "0"], ",", "m"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", SubscriptBox["z", "0"], "z"], RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[RowBox[List["m", " ", SuperscriptBox["t", "2"]]], "+", "1"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["t", "2"]]]]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Exists", "[", RowBox[List["\[Tau]", ",", " ", RowBox[List["{", RowBox[List[RowBox[List["\[Tau]", "\[Element]", "Reals"]], ",", " ", RowBox[List["0", "<", "\[Tau]", "<", "1"]]]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List[RowBox[List["m", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]]]], ")"]], "2"]]], "+", "1"]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["m", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]]]], ")"]], "2"]]], "+", "1"]], "<", "0"]], "\[And]", RowBox[List[RowBox[List["Im", "[", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]]]], ")"]], "2"]]]]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]]]], ")"]], "2"]]]]], "<", "0"]]]]]], "]"]], "]"]]]]]]

 MathML Form

 sd - 1 ( z m ) m z 2 + 1 cn ( sd - 1 ( z m ) m ) ( m - 1 ) z 2 + 1 z 0 z 1 m t 2 + 1 1 - ( 1 - m ) t 2 t + sd - 1 ( z 0 m ) /; ¬ τ , { τ TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] , 0 < τ < 1 } ( Im ( m ( τ ( z - z 0 ) + z 0 ) 2 + 1 ) 0 m ( τ ( z - z 0 ) + z 0 ) 2 + 1 < 0 Im ( 1 - ( 1 - m ) ( τ ( z - z 0 ) + z 0 ) 2 ) 0 1 - ( 1 - m ) ( τ ( z - z 0 ) + z 0 ) 2 < 0 ) Condition InverseJacobiSD z m m z 2 1 1 2 JacobiCN InverseJacobiSD z m m m -1 z 2 1 1 2 -1 t Subscript z 0 z 1 m t 2 1 1 2 1 -1 1 -1 m t 2 1 2 -1 InverseJacobiSD Subscript z 0 m τ τ 0 τ 1 m τ z -1 Subscript z 0 Subscript z 0 2 1 0 m τ z -1 Subscript z 0 Subscript z 0 2 1 0 1 -1 1 -1 m τ z -1 Subscript z 0 Subscript z 0 2 0 1 -1 1 -1 m τ z -1 Subscript z 0 Subscript z 0 2 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiSD", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", SubscriptBox["zz", "0"], "z"], RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[RowBox[List["m", " ", SuperscriptBox["t", "2"]]], "+", "1"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["t", "2"]]]]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]]], "/;", RowBox[List["!", RowBox[List[SubscriptBox["\[Exists]", RowBox[List["\[Tau]", ",", RowBox[List["{", RowBox[List[RowBox[List["\[Tau]", "\[Element]", "Reals"]], ",", RowBox[List["0", "<", "\[Tau]", "<", "1"]]]], "}"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List[RowBox[List["m", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]]], ")"]], "2"]]], "+", "1"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["m", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]]], ")"]], "2"]]], "+", "1"]], "<", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]]], ")"]], "2"]]]]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]]], ")"]], "2"]]]]], "<", "0"]]]], ")"]]]]]]]]]]]]

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

 2007-05-02