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variants of this functions
EllipticTheta




Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticTheta[4,z,q] > Series representations > q-series > Expansions at q==0





http://functions.wolfram.com/09.04.06.0022.01









  


  








Input Form





EllipticTheta[4, z, q] == 1 - 2 q Cos[2 z] + 2 q^4 Cos[4 z] - 2 q^9 Cos[6 z] + 2 q^16 Cos[8 z] + \[Ellipsis] /; (q -> 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List["1", "-", RowBox[List["2", " ", "q", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["q", "4"], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox["q", "9"], " ", RowBox[List["Cos", "[", RowBox[List["6", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["q", "16"], " ", RowBox[List["Cos", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", "\[Ellipsis]"]]]], "/;", " ", RowBox[List["(", RowBox[List["q", "\[Rule]", "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> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> cos </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mn> 16 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <ci> z </ci> <ci> q </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> cos </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <ci> q </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 16 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> q </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "q", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["q", "4"], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox["q", "9"], " ", RowBox[List["Cos", "[", RowBox[List["6", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["q", "16"], " ", RowBox[List["Cos", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["q", "\[Rule]", "0"]], ")"]]]]]]]]


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





2007-05-02