Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











JacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Identities involving the group of functions > Higher order local identities > Local identities of arbitrary rank





http://functions.wolfram.com/09.29.18.0147.01









  


  










Input Form





m^n JacobiSN[z, m]^(2 n) JacobiCN[z, m] JacobiDN[z + a, m] == JacobiNS[a, m]^(2 n) JacobiDS[a, m] JacobiSN[z + a, m] - m^n JacobiCS[a, m] JacobiSN[z, m]^(2 n + 1) - JacobiDS[a, m] (JacobiCS[a, m] JacobiDS[a, m] JacobiSN[z, m] + JacobiNS[a, m] JacobiCN[z, m] JacobiDN[z, m]) Sum[m^(n - k - 1) JacobiNS[a, m]^(2 k) JacobiSN[z, m]^(2 (n - k - 1)), {k, 0, n - 1}] /; Element[n, Integers] && n - 1 >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["m", "n"], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["2", "n"]]], RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", "n"]]], RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "-", RowBox[List[SuperscriptBox["m", "n"], RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]]]], "-", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "k", "-", "1"]]], SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", "k"]]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["2", RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["n", "-", "1"]], "\[GreaterEqual]", "0"]]]]]]]]










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> <msup> <mi> m </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cn </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> <mrow> <mo> - </mo> <msup> <mi> m </mi> <mi> n </mi> </msup> </mrow> <mo> &#8290; </mo> <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> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mi> ds </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> ns </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sn </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> - </mo> <mrow> <mrow> <mi> ds </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> <mrow> <mi> cn </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> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> ns </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </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> <mi> ds </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mi> m </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> ns </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <power /> <ci> m </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> JacobiCN </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> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["m_", "n_"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], RowBox[List["2", " ", "n_"]]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "-", RowBox[List[SuperscriptBox["m", "n"], " ", RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]], "-", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "k", "-", "1"]]], " ", SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", " ", "k"]]], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "1"]], "\[GreaterEqual]", "0"]]]]]]]]]]










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