Jacobi elliptic function ns: Complex characteristics (formula 09.33.19.0003)

JacobiNS

$Mathematica Notation$

Elliptic Functions

JacobiNS[z,m]

Complex characteristics

Absolute value

http://functions.wolfram.com/09.33.19.0003.01

Input Form

Abs[JacobiNS[x + I y, m]] == Sqrt[(JacobiDN[y, 1 - m]^2 JacobiSN[x, m]^2 + JacobiCN[x, m]^2 JacobiCN[y, 1 - m]^2 JacobiDN[x, m]^2 JacobiSN[y, 1 - m]^2) (JacobiCN[y, 1 - m]^2 + m JacobiSN[x, m]^2 JacobiSN[y, 1 - m]^2)^2]/ (JacobiDN[y, 1 - m]^2 JacobiSN[x, m]^2 + JacobiCN[x, m]^2 JacobiCN[y, 1 - m]^2 JacobiDN[x, m]^2 JacobiSN[y, 1 - m]^2) /; Element[{x, y, m}, Reals]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ",", "m"]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], "+", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"]]]]], ")"]], "2"]]], ")"]]]], "/", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["{", RowBox[List["x", ",", "y", ",", "m"]], "}"]], "\[Element]", "Reals"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mo> √ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <abs /> <apply> <ci> JacobiNS </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <root /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> x </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> JacobiCN </ci> <ci> x </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiCN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> x </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> JacobiCN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> x </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> x </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> JacobiCN </ci> <ci> x </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiCN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> x </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <list> <ci> x </ci> <ci> y </ci> <ci> m </ci> </list> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], ",", "m_"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], "+", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"]]]]], ")"]], "2"]]]], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], "2"]]]]]], "/;", RowBox[List[RowBox[List["{", RowBox[List["x", ",", "y", ",", "m"]], "}"]], "\[Element]", "Reals"]]]]]]]]

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

2007-05-02