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variants of this functions
EllipticTheta




Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticTheta[1,z,q] > Representations through equivalent functions > With related functions > Involving Weierstrass functions





http://functions.wolfram.com/09.01.27.0010.01









  


  








Input Form





EllipticTheta[1, z, q] == (Pi/Subscript[\[Omega], 1]) q^(1/4) Exp[-((2 Subscript[\[Eta], 1] Subscript[\[Omega], 1] z^2)/Pi^2)] Product[1 - q^(2 n), {n, 1, Infinity}]^3 WeierstrassSigma[ (2 Subscript[\[Omega], 1] z)/Pi, {Subscript[g, 2], Subscript[g, 3]}] /; {Subscript[\[Omega], 1], Subscript[\[Omega], 3]} == WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}] && Subscript[\[Eta], 1] == WeierstrassZeta[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}] && q == Exp[Pi I (Subscript[\[Omega], 3]/Subscript[\[Omega], 1])]








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["q", RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["2", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"], " ", SuperscriptBox["z", "2"]]], SuperscriptBox["\[Pi]", "2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "n"]]]]], ")"]]]], ")"]], "3"], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List[FractionBox[RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"], " ", "z"]], "\[Pi]"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], SubscriptBox["\[Omega]", "1"]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]], "&&", RowBox[List[SubscriptBox["\[Eta]", "1"], "\[Equal]", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "&&", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SubscriptBox["\[Omega]", "3"]]], SubscriptBox["\[Omega]", "1"]]]]]]]]]]]]]


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





2001-10-29