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http://functions.wolfram.com/09.15.16.0002.01
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WeierstrassSigma[Subscript[z, 1] + Subscript[z, 2],
{Subscript[g, 2], Subscript[g, 3]}] ==
Exp[Subscript[z, 1] WeierstrassZeta[Subscript[z, 2],
{Subscript[g, 2], Subscript[g, 3]}] - (Subscript[z, 1]^2/2)
WeierstrassP[Subscript[z, 2], {Subscript[g, 2], Subscript[g, 3]}]]
WeierstrassSigma[Subscript[z, 2], {Subscript[g, 2], Subscript[g, 3]}]
Product[(1 - Subscript[z, 1]/(2 m Subscript[\[Omega], 1] +
2 n Subscript[\[Omega], 3] - Subscript[z, 2]))
Exp[Subscript[z, 1]/(2 m Subscript[\[Omega], 1] +
2 n Subscript[\[Omega], 3] - Subscript[z, 2]) +
(1/2) (Subscript[z, 1]^2/(2 m Subscript[\[Omega], 1] +
2 n Subscript[\[Omega], 3] - Subscript[z, 2])^2)],
{m, -Infinity, Infinity}, {n, -Infinity, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", "\[Equal]", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List[SubscriptBox["z", "1"], " ", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["z", "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "-", RowBox[List[FractionBox[SubsuperscriptBox["z", "1", "2"], "2"], " ", RowBox[List["WeierstrassP", "[", RowBox[List[SubscriptBox["z", "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]], "]"]], RowBox[List["WeierstrassSigma", "[", RowBox[List[SubscriptBox["z", "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox[SubscriptBox["z", "1"], RowBox[List[RowBox[List["2", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", "n", " ", SubscriptBox["\[Omega]", "3"]]], "-", SubscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["Exp", "[", RowBox[List[FractionBox[SubscriptBox["z", "1"], RowBox[List[RowBox[List["2", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", "n", " ", SubscriptBox["\[Omega]", "3"]]], "-", SubscriptBox["z", "2"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", FractionBox[SubsuperscriptBox["z", "1", "2"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", "n", " ", SubscriptBox["\[Omega]", "3"]]], "-", SubscriptBox["z", "2"]]], ")"]], "2"]]]]]], "]"]]]]]]]]]]]]]]
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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> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </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[SubscriptBox["z", "2"], "+", SubscriptBox["z", "1"]]], 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> <mtext> </mtext> <mo> ⩵ </mo> <mrow> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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["\[Zeta]", "(", RowBox[List[RowBox[List[TagBox[SubscriptBox["z", "2"], Rule[Editable, True]], ";", TagBox[SubscriptBox["g", "2"], Rule[Editable, True]]]], ",", TagBox[SubscriptBox["g", "3"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mtext> </mtext> <mrow> <mfrac> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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[SubscriptBox["z", "2"], 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> <munderover> <mo> ∏ </mo> <mrow> <mi> m </mi> <mo> , </mo> <mrow> <mi> n </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mfrac> <mo> + </mo> <mfrac> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <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> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </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> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> exp </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ζ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </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> WeierstrassZeta </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> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> SubsuperscriptBox </ci> <ms> z </ms> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ℘ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </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> SubscriptBox </ci> <ms> z </ms> <ms> 2 </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> <ci> RowBox </ci> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> ∞ </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ∞ </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </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> n </ms> <apply> <ci> SubscriptBox </ci> <ms> ω </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> exp </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </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> n </ms> <apply> <ci> SubscriptBox </ci> <ms> ω </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubsuperscriptBox </ci> <ms> z </ms> <ms> 1 </ms> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </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> n </ms> <apply> <ci> SubscriptBox </ci> <ms> ω </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </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[SubscriptBox["z_", "1"], "+", SubscriptBox["z_", "2"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[SubscriptBox["zz", "1"], " ", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["zz", "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SubsuperscriptBox["zz", "1", "2"], " ", RowBox[List["WeierstrassP", "[", RowBox[List[SubscriptBox["zz", "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]]]]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List[SubscriptBox["zz", "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox[SubscriptBox["zz", "1"], RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "3"]]], "-", SubscriptBox["zz", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SubscriptBox["zz", "1"], RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "3"]]], "-", SubscriptBox["zz", "2"]]]], "+", FractionBox[SubsuperscriptBox["zz", "1", "2"], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "3"]]], "-", SubscriptBox["zz", "2"]]], ")"]], "2"]]]]]]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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