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variants of this functions
EllipticTheta




Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticTheta[2,z,q] > Series representations > q-series > Expansions at q==0





http://functions.wolfram.com/09.02.06.0002.01









  


  








Input Form





EllipticTheta[2, z, q] == q^(1/4) Sum[q^(k (k + 1)) E^((2 k + 1) I z), {k, -Infinity, Infinity}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["q", RowBox[List["1", "/", "4"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], " ", "\[ImaginaryI]", " ", "z"]]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mroot> <mi> q </mi> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mi> q </mi> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> q </ci> </apply> <apply> <times /> <apply> <power /> <ci> q </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> q </ci> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["q", RowBox[List["1", "/", "4"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "\[ImaginaryI]", " ", "z"]]]]]]]]]]]]]


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





2001-10-29