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 WeierstrassSigma

 http://functions.wolfram.com/09.15.16.0019.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Exp", "[", RowBox[List[FractionBox[SuperscriptBox["z", "2"], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["2", "n"]]], RowBox[List["WeierstrassP", "[", RowBox[List[FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]], "]"]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["WeierstrassSigma", "[", RowBox[List[FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "2"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 σ ( z ; g 2 ( ω 1 2 n + 1 , ω 3 ) , g 3 ( ω 1 2 n + 1 , ω 3 ) ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", TagBox[RowBox[List[SubscriptBox["g", "2"], "(", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], ",", SubscriptBox["\[Omega]", "3"]]], ")"]], Rule[Editable, True]]]], ",", RowBox[List[SubscriptBox["g", "3"], "(", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], ",", SubscriptBox["\[Omega]", "3"]]], ")"]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] ( - 1 ) n exp ( z 2 2 k = 1 2 n ( 2 k ω 1 2 n + 1 ; g 2 , g 3 ) ) σ ( z ; g 2 , g 3 ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] k = 1 n ( σ ( z + 2 k ω 1 2 n + 1 ; g 2 , g 3 ) σ ( z - 2 k ω 1 2 n + 1 ; g 2 , g 3 ) TagBox[RowBox[List[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["z", "+", FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]]]], Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], " ", RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["z", "-", FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]]]], Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]]]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] / σ ( 2 k ω 1 2 n + 1 ; g 2 , g 3 ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + σ ( z ; g 2 ( ω 1 2 n + 1 , ω 3 ) , g 3 ( ω 1 2 n + 1 , ω 3 ) ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", TagBox[RowBox[List[SubscriptBox["g", "2"], "(", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], ",", SubscriptBox["\[Omega]", "3"]]], ")"]], Rule[Editable, True]]]], ",", RowBox[List[SubscriptBox["g", "3"], "(", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], ",", SubscriptBox["\[Omega]", "3"]]], ")"]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] ( - 1 ) n exp ( z 2 2 k = 1 2 n ( 2 k ω 1 2 n + 1 ; g 2 , g 3 ) ) σ ( z ; g 2 , g 3 ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] k = 1 n ( σ ( z + 2 k ω 1 2 n + 1 ; g 2 , g 3 ) σ ( z - 2 k ω 1 2 n + 1 ; g 2 , g 3 ) TagBox[RowBox[List[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["z", "+", FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]]]], Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], " ", RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["z", "-", FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]]]], Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]]]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] / σ ( 2 k ω 1 2 n + 1 ; g 2 , g 3 ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[FractionBox[RowBox[List["2", "k", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], Rule[Editable, True]], ";", SubscriptBox["g", "2"]]], ",", SubscriptBox["g", "3"]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassSigma", "[", RowBox[List["z_", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["\[Omega]_", "1"], RowBox[List[RowBox[List["2", " ", "n_"]], "+", "1"]]], ",", SubscriptBox["\[Omega]_", "3"]]], "}"]], "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["2", " ", "n"]]], RowBox[List["WeierstrassP", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], SuperscriptBox[RowBox[List["WeierstrassSigma", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "k", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "2"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2001-10-29