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http://functions.wolfram.com/09.16.16.0007.01
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WeierstrassSigma[k, Subscript[z, 1] + Subscript[z, 2],
{Subscript[g, 2], Subscript[g, 3]}] WeierstrassSigma[j,
Subscript[z, 1] - Subscript[z, 2], {Subscript[g, 2], Subscript[g, 3]}] ==
WeierstrassSigma[k, Subscript[z, 1], {Subscript[g, 2], Subscript[g, 3]}]
WeierstrassSigma[j, Subscript[z, 1], {Subscript[g, 2], Subscript[g, 3]}]
WeierstrassSigma[k, Subscript[z, 2], {Subscript[g, 2], Subscript[g, 3]}]
WeierstrassSigma[j, Subscript[z, 2], {Subscript[g, 2],
Subscript[g, 3]}] + (Subscript[e, j] - Subscript[e, k])
WeierstrassSigma[Subscript[z, 1], {Subscript[g, 2], Subscript[g, 3]}]
WeierstrassSigma[i, Subscript[z, 1], {Subscript[g, 2], Subscript[g, 3]}]
WeierstrassSigma[Subscript[z, 2], {Subscript[g, 2], Subscript[g, 3]}]
WeierstrassSigma[i, Subscript[z, 2], {Subscript[g, 2],
Subscript[g, 3]}] /; Element[{i, j, k}, {1, 2, 3}] && i != j != k
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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> <mrow> <msub> <mi> σ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]]], Rule[Editable, True]] </annotation> </semantics> <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> <semantics> <mrow> <msub> <mi> σ </mi> <mi> j </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </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[SubscriptBox["\[Sigma]", "j"], "(", RowBox[List[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "1"], "-", 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> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <semantics> <mrow> <msub> <mi> σ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </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[SubscriptBox["\[Sigma]", "k"], "(", RowBox[List[RowBox[List[TagBox[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> <mo> ⁢ </mo> <mrow> <msub> <mi> σ </mi> <mi> j </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "1"], Rule[Editable, True]] </annotation> </semantics> <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> <msub> <mi> σ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "2"], Rule[Editable, True]] </annotation> </semantics> <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> <msub> <mi> σ </mi> <mi> j </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "2"], Rule[Editable, True]] </annotation> </semantics> <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> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> e </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> e </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </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[SubscriptBox["z", "1"], ";", 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> <msub> <mi> σ </mi> <mi> i </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "1"], Rule[Editable, True]] </annotation> </semantics> <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> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> <msub> <mi> σ </mi> <mi> i </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> , </mo> <mi> k </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> <mo> ≠ </mo> <mi> k </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <msub> <mi> σ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]]], Rule[Editable, True]] </annotation> </semantics> <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> <semantics> <mrow> <msub> <mi> σ </mi> <mi> j </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </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[SubscriptBox["\[Sigma]", "j"], "(", RowBox[List[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "1"], "-", 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> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <semantics> <mrow> <msub> <mi> σ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </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[SubscriptBox["\[Sigma]", "k"], "(", RowBox[List[RowBox[List[TagBox[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> <mo> ⁢ </mo> <mrow> <msub> <mi> σ </mi> <mi> j </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "1"], Rule[Editable, True]] </annotation> </semantics> <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> <msub> <mi> σ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "2"], Rule[Editable, True]] </annotation> </semantics> <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> <msub> <mi> σ </mi> <mi> j </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "2"], Rule[Editable, True]] </annotation> </semantics> <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> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> e </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> e </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </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[SubscriptBox["z", "1"], ";", 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> <msub> <mi> σ </mi> <mi> i </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["z", "1"], Rule[Editable, True]] </annotation> </semantics> <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> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> <msub> <mi> σ </mi> <mi> i </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> , </mo> <mi> k </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> <mo> ≠ </mo> <mi> k </mi> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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