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variants of this functions
WeierstrassSigma






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassSigma[n,z,{g2,g3}] > Differentiation > Symbolic differentiation > With respect to z





http://functions.wolfram.com/09.16.20.0003.01









  


  










Input Form





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





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MathML Form







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</ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










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["m_", ",", "z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["\[Eta]", "m"]]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["m", ",", "z", ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]], "+", FractionBox[RowBox[List[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["\[ExponentialE]", RowBox[List[RowBox[List["-", SubscriptBox["\[Eta]", "m"]]], " ", "z"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]]]]], ")"]], "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[RowBox[List["(", RowBox[List["z", "+", SubscriptBox["\[Omega]", "m"]]], ")"]], "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]", " ", RowBox[List["(", RowBox[List["z", "+", SubscriptBox["\[Omega]", "m"]]], ")"]]]]], ")"]], "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[RowBox[List["(", RowBox[List["z", "+", SubscriptBox["\[Omega]", "m"]]], ")"]], " ", 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["WeierstrassSigma", "[", RowBox[List[SubscriptBox["\[Omega]", "m"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.