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
 











variants of this functions
EllipticTheta




Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticTheta[1,z,q] > Identities involving the group of functions > Basic Algebraic Identities > Alternative version of fundamental identities (from Tannery and Molk)





http://functions.wolfram.com/09.01.18.0080.01









  


  








Input Form





EllipticTheta[3, a + b, q] EllipticTheta[3, a - b, q] EllipticTheta[3, c + d, q] EllipticTheta[3, c - d, q] + EllipticTheta[4, a + b, q] EllipticTheta[4, a - b, q] EllipticTheta[4, c + d, q] EllipticTheta[4, c - d, q] == EllipticTheta[3, a + d, q] EllipticTheta[3, a - d, q] EllipticTheta[3, b + c, q] EllipticTheta[3, b - c, q] + EllipticTheta[4, a + d, q] EllipticTheta[4, a - d, q] EllipticTheta[4, b + c, q] EllipticTheta[4, b - c, q]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "+", "b"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "-", "b"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["c", "+", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["c", "-", "d"]], ",", "q"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a", "+", "b"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a", "-", "b"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["c", "+", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["c", "-", "d"]], ",", "q"]], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "+", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "-", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["b", "+", "c"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["b", "-", "c"]], ",", "q"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a", "+", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a", "-", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["b", "+", "c"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["b", "-", "c"]], ",", "q"]], "]"]]]]]]]]]]








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> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> c </ci> <ci> d </ci> </apply> <ci> q </ci> </apply> </apply> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> c </ci> <ci> d </ci> </apply> <ci> q </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> a </ci> <ci> d </ci> </apply> <ci> q </ci> </apply> </apply> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> a </ci> <ci> d </ci> </apply> <ci> q </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a_", "+", "b_"]], ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a_", "-", "b_"]], ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["c_", "+", "d_"]], ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["c_", "-", "d_"]], ",", "q_"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a_", "+", "b_"]], ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a_", "-", "b_"]], ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["c_", "+", "d_"]], ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["c_", "-", "d_"]], ",", "q_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "+", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "-", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["b", "+", "c"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["b", "-", "c"]], ",", "q"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a", "+", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["a", "-", "d"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["b", "+", "c"]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["b", "-", "c"]], ",", "q"]], "]"]]]]]]]]]]


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





2001-10-29