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 > The 16 fundamental algebraic identities (from Enneper)





http://functions.wolfram.com/09.01.18.0054.01









  


  








Input Form





EllipticTheta[3, a, q] EllipticTheta[3, b, q] EllipticTheta[3, c, q] EllipticTheta[3, d, q] - EllipticTheta[2, a, q] EllipticTheta[2, b, q] EllipticTheta[2, c, q] EllipticTheta[2, d, q] == EllipticTheta[4, (1/2) (a + b + c + d), q] EllipticTheta[4, (1/2) (a + b - c - d), q] EllipticTheta[4, (1/2) (a - b + c - d), q] EllipticTheta[4, (1/2) (a - b - c + d), q] + EllipticTheta[1, (1/2) (a + b + c + d), q] EllipticTheta[1, (1/2) (a + b - c - d), q] EllipticTheta[1, (1/2) (a - b + c - d), q] EllipticTheta[1, (1/2) (a - b - c + d), q]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "a", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "b", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "c", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "d", ",", "q"]], "]"]]]], "-", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "a", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "b", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "c", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "d", ",", "q"]], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List["a", "+", "b", "+", "c", "+", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", "c", "-", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", "c", "-", "d"]], ")"]]]], " ", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "-", "c", "+", "d"]], ")"]]]], ",", "q"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List["a", "+", "b", "+", "c", "+", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", "c", "-", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", "c", "-", "d"]], ")"]]]], " ", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "-", "c", "+", "d"]], ")"]]]], ",", "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> <mi> a </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> c </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> d </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> c </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> d </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </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> <ci> a </ci> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <ci> b </ci> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <ci> c </ci> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <ci> d </ci> <ci> q </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <ci> d </ci> <ci> q </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> d </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> d </ci> </apply> </apply> <ci> q </ci> </apply> </apply> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> d </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> d </ci> </apply> </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", ",", "a_", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "b_", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "c_", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "d_", ",", "q_"]], "]"]]]], "-", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "a_", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "b_", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "c_", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "d_", ",", "q_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", "c", "+", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", "c", "-", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", "c", "-", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "-", "c", "+", "d"]], ")"]]]], ",", "q"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", "c", "+", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", "c", "-", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", "c", "-", "d"]], ")"]]]], ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "-", "c", "+", "d"]], ")"]]]], ",", "q"]], "]"]]]]]]]]]]


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





2001-10-29