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variants of this functions
SiegelTheta






Mathematica Notation

Traditional Notation









Elliptic Functions > SiegelTheta[{{u1,...,ur},{v1,...,vr}},{{m1,1,...,m1,r},...,{mr,1,...,mr,r}},{s1,...,sr}] > Primary definition





http://functions.wolfram.com/09.59.02.0001.01









  


  










Input Form





SiegelTheta[{{Subscript[u, 1], \[Ellipsis], Subscript[u, r]}, {Subscript[v, 1], \[Ellipsis], Subscript[v, r]}}, {{Subscript[m, 1, 1], \[Ellipsis], Subscript[m, 1, r]}, \[Ellipsis], {Subscript[m, r, 1], \[Ellipsis], Subscript[m, r, r]}}, {Subscript[s, 1], \[Ellipsis], Subscript[s, r]}] == Sum[\[Ellipsis] Sum[Exp[I Pi ((n + u) \[CenterDot] \[CapitalOmega] \[CenterDot] (n + u) + 2 (n + u) \[CenterDot] (s + v))], {Subscript[n, r], -Infinity, Infinity}], {Subscript[n, 1], -Infinity, Infinity}] /; u = {Subscript[u, 1], \[Ellipsis], Subscript[u, r]} && v == {Subscript[v, 1], \[Ellipsis], Subscript[v, r]} && \[CapitalOmega] == {{Subscript[m, 1, 1], \[Ellipsis], Subscript[m, 1, r]}, \[Ellipsis], {Subscript[m, r, 1], \[Ellipsis], Subscript[m, r, r]}} && s == {Subscript[s, 1], \[Ellipsis], Subscript[s, r]} && n = {Subscript[n, 1], \[Ellipsis], Subscript[n, r]} && n + u == {Subscript[n, 1] + Subscript[u, 1], \[Ellipsis], Subscript[n, r] + Subscript[u, r]} && s + v == {Subscript[s, 1] + Subscript[v, 1], \[Ellipsis], Subscript[s, r] + Subscript[v, r]}










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["SiegelTheta", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["u", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["u", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["v", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["v", "r"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["m", RowBox[List["1", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["m", RowBox[List["1", ",", "r"]]]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["m", RowBox[List["r", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["m", RowBox[List["r", ",", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["s", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["s", "r"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["n", "1"], "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["n", "r"], "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["Exp", "[", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "u"]], ")"]], "\[CenterDot]", "\[CapitalOmega]", "\[CenterDot]", RowBox[List["(", RowBox[List["n", "+", "u"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "u"]], ")"]], "\[CenterDot]", RowBox[List["(", RowBox[List["s", "+", "v"]], ")"]]]]]]]], ")"]]]], "]"]]]]]]]]]], "/;", "u"]], "=", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["u", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["u", "r"]]], "}"]], "\[And]", RowBox[List["v", "\[Equal]", RowBox[List["{", RowBox[List[SubscriptBox["v", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["v", "r"]]], "}"]]]], "\[And]", RowBox[List["\[CapitalOmega]", "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["m", RowBox[List["1", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["m", RowBox[List["1", ",", "r"]]]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["m", RowBox[List["r", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["m", RowBox[List["r", ",", "r"]]]]], "}"]]]], "}"]]]], "\[And]", RowBox[List["s", "\[Equal]", RowBox[List["{", RowBox[List[SubscriptBox["s", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["s", "r"]]], "}"]]]], "\[And]", "n"]], "=", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "r"]]], "}"]], "\[And]", RowBox[List[RowBox[List["n", "+", "u"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["u", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["n", "r"], "+", SubscriptBox["u", "r"]]]]], "}"]]]], "\[And]", RowBox[List[RowBox[List["s", "+", "v"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["s", "1"], "+", SubscriptBox["v", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["s", "r"], "+", SubscriptBox["v", "r"]]]]], "}"]]]]]]]]]]]]










MathML Form







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</mi> </munderover> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#183; </mo> <mi> &#937; </mi> <mo> &#183; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#183; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mi> v </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> u </mi> </mrow> <mo> = </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> u </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> u </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> v </mi> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> v </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> v </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#937; </mi> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> r </mi> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> , </mo> <mi> r </mi> </mrow> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> s </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> s </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> u </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> n </mi> <mi> r </mi> </msub> <mo> + </mo> <msub> <mi> u </mi> <mi> r </mi> </msub> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> s </mi> <mo> + </mo> <mi> v </mi> </mrow> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> s </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> v </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> s </mi> <mi> r </mi> </msub> <mo> + </mo> <msub> <mi> v </mi> <mi> r </mi> </msub> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <mrow> <mi> &#920; </mi> <mo> [ </mo> <mtext> &#8288; </mtext> <mtable> <mtr> <mtd> <mrow> <mo> { </mo> <mrow> <msub> <mi> u </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> u </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> { </mo> <mrow> <msub> <mi> v </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> v </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> </mtd> </mtr> </mtable> <mtext> &#8288; </mtext> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <mo> &#8230; </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> r </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> &#8230; </mo> </mtd> <mtd> <mo> &#8230; </mo> </mtd> <mtd> <mo> &#8230; </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <mo> &#8230; </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> , </mo> <mi> r </mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> s </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> s </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> n </mi> <mi> r </mi> </msub> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#183; </mo> <mi> &#937; </mi> <mo> &#183; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#183; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mi> v </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> u </mi> </mrow> <mo> = </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> u </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> u </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> v </mi> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> v </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> v </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#937; </mi> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> r </mi> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> , </mo> <mi> r </mi> </mrow> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> s </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> s </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> r </mi> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> u </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> n </mi> <mi> r </mi> </msub> <mo> + </mo> <msub> <mi> u </mi> <mi> r </mi> </msub> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> s </mi> <mo> + </mo> <mi> v </mi> </mrow> <mo> &#63449; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> s </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> v </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> s </mi> <mi> r </mi> </msub> <mo> + </mo> <msub> <mi> v </mi> <mi> r </mi> </msub> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





© 1998- Wolfram Research, Inc.