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 WeierstrassSigma

 http://functions.wolfram.com/09.15.20.0011.01

 Input Form

 D[WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}], {z, n}] == (2 Subscript[\[Omega], 1])^(1 - n) Pi^(n - 1/2) Product[1/(1 - q^(2 m)), {m, 1, Infinity}]^3 Sum[HypergeometricPFQRegularized[{1/2, 1}, {(1 - j)/2, (2 - j)/2}, (z^2/(2 Subscript[\[Omega], 1])) WeierstrassZeta[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}]] ((4 Subscript[\[Omega], 1])/(Pi z))^j Binomial[n, j] Sum[(-1)^k q^(k (k + 1)) (1 + 2 k)^(n - j) Sin[(Pi (z (1 + 2 k) + (n - j) Subscript[\[Omega], 1]))/ (2 Subscript[\[Omega], 1])], {k, 0, Infinity}], {j, 0, n}] /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", SubscriptBox["\[Omega]", "1"]]], ")"]], RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["n", "-", FractionBox["1", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", "m"]]]]]]]], ")"]], "3"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", FractionBox[RowBox[List["2", "-", "j"]], "2"]]], "}"]], ",", RowBox[List[FractionBox[SuperscriptBox["z", "2"], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["4", SubscriptBox["\[Omega]", "1"]]], RowBox[List["\[Pi]", " ", "z"]]], ")"]], "j"], RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "-", "j"]]], RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["z", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], " ", SubscriptBox["\[Omega]", "1"]]]]], ")"]]]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], "]"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 n σ ( z ; g 2 , g 3 ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox["z", 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]]]]]] z n ( 2 ω 1 ) 1 - n π n - 1 2 ( m = 1 1 1 - q 2 m ) 3 j = 0 n 2 F ~ 2 ( 1 2 , 1 ; 1 - j 2 , 2 - j 2 ; z 2 ζ ( ω 1 ; g 2 , g 3 ) 2 ω 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "j"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "j"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["z", "2"], " ", TagBox[TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[RowBox[List[TagBox[SubscriptBox["\[Omega]", "1"], 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]]]]]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]]]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( 4 ω 1 π z ) j ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k = 0 ( - 1 ) k q k ( k + 1 ) ( 2 k + 1 ) n - j sin ( π ( ( 2 k + 1 ) z + ( n - j ) ω 1 ) 2 ω 1 ) /; n + n σ ( z ; g 2 , g 3 ) TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox["z", 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]]]]]] z n ( 2 ω 1 ) 1 - n π n - 1 2 ( m = 1 1 1 - q 2 m ) 3 j = 0 n 2 F ~ 2 ( 1 2 , 1 ; 1 - j 2 , 2 - j 2 ; z 2 ζ ( ω 1 ; g 2 , g 3 ) 2 ω 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "j"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "j"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["z", "2"], " ", TagBox[TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[RowBox[List[TagBox[SubscriptBox["\[Omega]", "1"], 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]]]]]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]]]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( 4 ω 1 π z ) j ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k = 0 ( - 1 ) k q k ( k + 1 ) ( 2 k + 1 ) n - j sin ( π ( ( 2 k + 1 ) z + ( n - j ) ω 1 ) 2 ω 1 ) /; n + [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["WeierstrassSigma", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]], ")"]], RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["n", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "m"]]]]]]]], ")"]], "3"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", FractionBox[RowBox[List["2", "-", "j"]], "2"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["4", " ", SubscriptBox["\[Omega]", "1"]]], RowBox[List["\[Pi]", " ", "z"]]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "-", "j"]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], " ", SubscriptBox["\[Omega]", "1"]]]]], ")"]]]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2001-10-29