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http://functions.wolfram.com/09.15.16.0014.01
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WeierstrassSigma[n z, {Subscript[g, 2], Subscript[g, 3]}] ==
(n (-1)^(n^2 - 1) WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}]
Product[If[{j, k} == {0, 0}, 1,
WeierstrassSigma[z - (2 j Subscript[\[Omega], 1] +
2 k Subscript[\[Omega], 3])/n, {Subscript[g, 2], Subscript[g, 3]}]/
WeierstrassSigma[(2 j Subscript[\[Omega], 1] +
2 k Subscript[\[Omega], 3])/n, {Subscript[g, 2],
Subscript[g, 3]}]], {j, 0, n - 1}, {k, 0, n - 1}])/
E^(n (n - 1) Subscript[\[Eta], 2] z) /; Element[n - 1, Integers] &&
n - 1 > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["n", " ", "z"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List["n", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SuperscriptBox["n", "2"], "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "n"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", SubscriptBox["\[Eta]", "2"], " ", "z"]]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]]]], ",", "1", ",", FractionBox[RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List[RowBox[List["2", "j", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "3"]]]]], "n"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], RowBox[List["WeierstrassSigma", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "j", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "3"]]]]], "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["n", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["n", "-", "1"]], ">", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["n", " ", "z"]], Rule[Editable, True]], ";", TagBox[SubscriptBox["g", "2"], Rule[Editable, True]]]], ",", TagBox[SubscriptBox["g", "3"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <semantics> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["g", "2"], Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> , </mo> <semantics> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["g", "3"], Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <munder> <mrow> <mi> j </mi> <mo> , </mo> <mtext> </mtext> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> </mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> ≠ </mo> <mrow> <mo> { </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </munder> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["z", "-", FractionBox[RowBox[List[RowBox[List["2", " ", "j", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]", "3"]]]]], "n"]]], Rule[Editable, True]], ";", TagBox[SubscriptBox["g", "2"], Rule[Editable, True]]]], ",", TagBox[SubscriptBox["g", "3"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["2", " ", "j", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]", "3"]]]]], "n"], Rule[Editable, True]], ";", TagBox[SubscriptBox["g", "2"], Rule[Editable, True]]]], ",", TagBox[SubscriptBox["g", "3"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> σ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> </ms> <ms> z </ms> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 2 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 3 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> WeierstrassSigma </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <list> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> </apply> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> n </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> ⅇ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> n </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> η </ms> <ms> 2 </ms> </apply> <ms> z </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> σ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 2 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 3 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> , </ms> <ms> k </ms> </list> </apply> <ms> } </ms> </list> </apply> <ms> ≠ </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> , </ms> <ms> 0 </ms> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> σ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> j </ms> <ms> </ms> <apply> <ci> SubscriptBox </ci> <ms> ω </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> k </ms> <ms> </ms> <apply> <ci> SubscriptBox </ci> <ms> ω </ms> <ms> 3 </ms> </apply> </list> </apply> </list> </apply> <ms> n </ms> </apply> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 2 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 3 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> WeierstrassSigma </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <list> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> σ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> j </ms> <ms> </ms> <apply> <ci> SubscriptBox </ci> <ms> ω </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> k </ms> <ms> </ms> <apply> <ci> SubscriptBox </ci> <ms> ω </ms> <ms> 3 </ms> </apply> </list> </apply> </list> </apply> <ms> n </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 2 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 3 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> WeierstrassSigma </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <list> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> </apply> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℕ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> <ms> + </ms> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["n_", " ", "z_"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["n", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SuperscriptBox["n", "2"], "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "n"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", SubscriptBox["\[Eta]", "2"], " ", "z"]]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]]]], ",", "1", ",", FractionBox[RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List[RowBox[List["2", " ", "j", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]", "3"]]]]], "n"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]], RowBox[List["WeierstrassSigma", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "j", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]", "3"]]]]], "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["n", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "1"]], ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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