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variants of this functions
WeierstrassSigma






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassSigma[n,z,{g2,g3}] > Product representations > Infinite products involving exponentials





http://functions.wolfram.com/09.16.08.0004.01









  


  










Input Form





WeierstrassSigma[i, z, {Subscript[g, 2], Subscript[g, 3]}] == Exp[(-(z^2/2)) Subscript[e, i]] Product[If[{m, n} == {0, 0}, 1, (1 - z/(2 m Subscript[\[Omega], 1] + 2 n Subscript[\[Omega], 2] - Subscript[\[Omega], i])) Exp[z/(2 m Subscript[\[Omega], 1] + 2 n Subscript[\[Omega], 2] - Subscript[\[Omega], i]) + (1/2) (z^2/(2 m Subscript[\[Omega], 1] + 2 n Subscript[\[Omega], 2] - Subscript[\[Omega], i])^2)]], {m, -Infinity, Infinity}, {n, -Infinity, Infinity}] /; Element[i, {1, 2, 3}]










Standard Form





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MathML Form







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</ms> <apply> <ci> RowBox </ci> <list> <ms> i </ms> <ms> &#8712; </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> , </ms> <ms> 2 </ms> <ms> , </ms> <ms> 3 </ms> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassSigma", "[", RowBox[List["i_", ",", "z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], " ", SubscriptBox["e", "i"]]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["m", ",", "n"]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]]]], ",", "1", ",", RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox["z", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "2"]]], "-", SubscriptBox["\[Omega]", "i"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["z", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "2"]]], "-", SubscriptBox["\[Omega]", "i"]]]], "+", FractionBox[SuperscriptBox["z", "2"], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "2"]]], "-", SubscriptBox["\[Omega]", "i"]]], ")"]], "2"]]]]]]]]]]], "]"]]]]]]]], "/;", RowBox[List["i", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]]]]]]]]










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





2001-10-29





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