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JacobiDN




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Identities involving the group of functions > Local identities of rank 2





http://functions.wolfram.com/09.29.18.0050.01









  


  








Input Form





JacobiDN[z, m] JacobiDN[z + a, m] == JacobiDN[a, m] + JacobiCS[a, m] (JacobiZeta[JacobiAmplitude[z + a, m], m] - JacobiZeta[JacobiAmplitude[z, m], m] - JacobiZeta[JacobiAmplitude[a, m], m])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["a", ",", "m"]], "]"]], "+", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["a", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]]]]]]]]]]








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> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> &#918; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#918; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#918; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <ci> JacobiDN </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <ci> a </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["a", ",", "m"]], "]"]], "+", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["a", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]]]]]]]]]]










References





A. Khare, A. Lakshminarayan, U. Sukhatme, "Local Identities Involving Jacobi Elliptic Functions", math-ph/0306028, (2003) http://arXiv.org/abs/math-ph/0306028


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





2003-08-21