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 InverseJacobiNC

 http://functions.wolfram.com/09.43.07.0003.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List[SubscriptBox["z", "0"], ",", "m"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], SqrtBox[RowBox[List["m", "+", SuperscriptBox["z", "2"], "-", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", SubscriptBox["z", "0"], "z"], RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["t", "2"], "-", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["t", "2"]]], "+", "m"]]]]]], 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[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[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[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"]]], "+", "m"]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[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"]]], "+", "m"]], "<", "0"]]]]]], "]"]], "]"]]]]]]

 MathML Form

 nc - 1 ( z m ) nc - 1 ( z 0 m ) + z 2 - 1 ds ( nc - 1 ( z m ) m ) - m z 2 + z 2 + m z 0 z 1 t 2 - 1 ( 1 - m ) t 2 + m t /; ¬ τ , { τ TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] , 0 < τ < 1 } ( Im ( ( τ ( z - z 0 ) + z 0 ) 2 - 1 ) 0 ( τ ( z - z 0 ) + z 0 ) 2 - 1 < 0 Im ( ( 1 - m ) ( τ ( z - z 0 ) + z 0 ) 2 + m ) 0 ( 1 - m ) ( τ ( z - z 0 ) + z 0 ) 2 + m < 0 ) Condition InverseJacobiNC z m InverseJacobiNC Subscript z 0 m z 2 -1 1 2 JacobiDS InverseJacobiNC z m m -1 m z 2 z 2 m 1 2 -1 t Subscript z 0 z 1 t 2 -1 1 2 1 -1 m t 2 m 1 2 -1 τ τ 0 τ 1 τ z -1 Subscript z 0 Subscript z 0 2 -1 0 τ z -1 Subscript z 0 Subscript z 0 2 -1 0 1 -1 m τ z -1 Subscript z 0 Subscript z 0 2 m 0 1 -1 m τ z -1 Subscript z 0 Subscript z 0 2 m 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiNC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", SubscriptBox["zz", "0"], "z"], RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["t", "2"], "-", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["t", "2"]]], "+", "m"]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], SqrtBox[RowBox[List["m", "+", SuperscriptBox["z", "2"], "-", RowBox[List["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[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[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[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"]]], "+", "m"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[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"]]], "+", "m"]], "<", "0"]]]], ")"]]]]]]]]]]]]

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

 2007-05-02