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variants of this functions
WeierstrassSigma






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassSigma[z,{g2,g3}] > Transformations > Related transformations > General fractions of half-periods





http://functions.wolfram.com/09.15.16.0018.01









  


  










Input Form





WeierstrassSigma[z, WeierstrassInvariants[{Subscript[\[Omega], 1]/n, Subscript[\[Omega], 3]}]] == Exp[(z^2/2) Sum[WeierstrassP[(j/n) 2 Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}], {j, 1, n - 1}] - z Sum[WeierstrassZeta[(j/n) 2 Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}], {j, 1, n - 1}]] WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}] Product[WeierstrassSigma[z + (j/n) 2 Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}]/WeierstrassSigma[ (j/n) 2 Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}], {j, 1, n - 1}] /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassSigma", "[", RowBox[List["z_", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["\[Omega]_", "1"], "n_"], ",", SubscriptBox["\[Omega]_", "3"]]], "}"]], "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List["WeierstrassP", "[", RowBox[List[FractionBox[RowBox[List["j", " ", "2", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]], "-", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List["WeierstrassZeta", "[", RowBox[List[FractionBox[RowBox[List["j", " ", "2", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]]]]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["j", " ", "2", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "n"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], RowBox[List["WeierstrassSigma", "[", RowBox[List[FractionBox[RowBox[List["j", " ", "2", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.