Jacobi elliptic function nc: Specific values (formula 09.31.03.0025)

JacobiNC

$Mathematica Notation$

Values at half-quarter-period points

http://functions.wolfram.com/09.31.03.0025.01

Input Form

JacobiNC[EllipticK[m]/2 + (I EllipticK[1 - m])/2, m] == (m^(1/4)/(1 - m)^(1/4)) ((1 + I)/Sqrt[2])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["JacobiNC", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]], "2"]]], ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[" ", RadicalBox["m", "4"]]], RadicalBox[RowBox[List["1", "-", "m"]], "4"]], " ", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], SqrtBox["2"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> nc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> K </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mroot> <mi> m </mi> <mn> 4 </mn> </mroot> </mrow> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mn> 4 </mn> </mroot> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiNC </ci> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> K </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiNC", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticK", "[", "m_", "]"]], "2"], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m_"]], "]"]]]]]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox["2"]]]]]]]]

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

2001-10-29