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http://functions.wolfram.com/09.13.06.0011.01
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WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] ==
-(Subscript[\[Eta], i]/Subscript[\[Omega], i]) +
(Pi^2/(4 Subscript[\[Omega], i]^2))
Sum[Csc[Pi ((z - 2 n Subscript[\[Omega], j])/(2 Subscript[\[Omega], i]))]^
2, {n, -Infinity, Infinity}] /; Element[{i, j}, {1, 2, 3}] && i != j
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["\[Eta]", "i"], SubscriptBox["\[Omega]", "i"]]]], "+", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["4", SubsuperscriptBox["\[Omega]", "i", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", FractionBox[RowBox[List["z", "-", RowBox[List["2", "n", " ", SubscriptBox["\[Omega]", "j"]]]]], RowBox[List["2", SubscriptBox["\[Omega]", "i"]]]]]], "]"]], "2"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["i", ",", "j"]], "}"]], "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]], "\[And]", RowBox[List["i", "\[NotEqual]", "j"]]]]]]]]
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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> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <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> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <mrow> <mo> - </mo> <mfrac> <msub> <mi> η </mi> <mi> i </mi> </msub> <msub> <mi> ω </mi> <mi> i </mi> </msub> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msubsup> <mi> ω </mi> <mi> i </mi> <mn> 2 </mn> </msubsup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mi> i </mi> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> i </mi> <mo> ≠ </mo> <mi> j </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> WeierstrassP </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> η </ci> <ci> i </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <ci> i </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <apply> <csc /> <apply> <times /> <pi /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> ω </ci> <ci> i </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <list> <ci> i </ci> <ci> j </ci> </list> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <cn type='integer'> 3 </cn> </list> </apply> <apply> <neq /> <ci> i </ci> <ci> j </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassP", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["\[Eta]", "i"], SubscriptBox["\[Omega]", "i"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], SuperscriptBox[RowBox[List["Csc", "[", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "j"]]]]], ")"]]]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "i"]]]], "]"]], "2"]]]]], RowBox[List["4", " ", SubsuperscriptBox["\[Omega]", "i", "2"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["i", ",", "j"]], "}"]], "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]], "&&", RowBox[List["i", "\[NotEqual]", "j"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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