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variants of this functions
WeierstrassSigma






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassSigma[z,{g2,g3}] > Integration > Indefinite integration > Involving only one direct function





http://functions.wolfram.com/09.15.21.0001.01









  


  










Input Form





Integrate[WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}], z] == (Subscript[\[Omega], 1]^(3/2)/Sqrt[2 Pi Subscript[\[Eta], 1]]) Product[1/(1 - q^(2 m)), {m, 1, Infinity}]^3 Sum[(-1)^k q^(k (k + 1)) Exp[(Pi^2 (2 k + 1)^2)/(8 Subscript[\[Eta], 1] Subscript[\[Omega], 1])] (Erf[(Pi (2 k + 1) + 2 I z Subscript[\[Eta], 1])/ (2 Sqrt[2 Subscript[\[Eta], 1] Subscript[\[Omega], 1]])] + Erf[(Pi (2 k + 1) - 2 I z Subscript[\[Eta], 1])/ (2 Sqrt[2 Subscript[\[Eta], 1] Subscript[\[Omega], 1]])]), {k, 0, Infinity}]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SubsuperscriptBox["\[Omega]", "1", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "m"]]]]]]]], ")"]], "3"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "2"]]], RowBox[List["8", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SubscriptBox["\[Eta]", "1"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"]]]]]]], "]"]], "+", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SubscriptBox["\[Eta]", "1"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"]]]]]]], "]"]]]], ")"]]]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]", " ", SubscriptBox["\[Eta]", "1"]]]]]]]]]










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





2001-10-29





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