Weierstrass zeta function: Transformations (formula 09.17.16.0013)

WeierstrassZeta

$Mathematica Notation$

Elliptic Functions

WeierstrassZeta[z,{g₂,g₃}]

Transformations

Related transformations

General fractions of half-periods

http://functions.wolfram.com/09.17.16.0013.01

Input Form

WeierstrassZeta[z, WeierstrassInvariants[{Subscript[\[Omega], 1]/n, Subscript[\[Omega], 3]}]] == WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}] + Sum[WeierstrassZeta[z + (2 k Subscript[\[Omega], 1])/n, {Subscript[g, 2], Subscript[g, 3]}] + z WeierstrassP[(2 k Subscript[\[Omega], 1])/n, {Subscript[g, 2], Subscript[g, 3]}] - (2 k Subscript[\[Eta], 1])/n, {k, 1, n - 1}] /; Element[n, Integers] && n > 0

Standard Form

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MathML Form

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</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["z", Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", Rule[Editable, True]] </annotation> </semantics> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </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> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </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> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 1 </mn> </msub> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <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'> <mrow> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mi> n </mi> </mfrac> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mi> n </mi> </mfrac> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", TagBox[RowBox[List[SubscriptBox["g", "2"], "(", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], "n"], ",", SubscriptBox["\[Omega]", "3"]]], ")"]], Rule[Editable, True]]]], ",", RowBox[List[SubscriptBox["g", "3"], "(", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], "n"], ",", SubscriptBox["\[Omega]", "3"]]], ")"]]]], ")"]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <semantics> <mrow> <mi> ζ </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> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", Rule[Editable, True]] </annotation> </semantics> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </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> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mi> n </mi> </mfrac> <mo> ; 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Rule Form

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

2001-10-29