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WeierstrassZeta






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.17.20.0009.01









  


  










Input Form





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










Standard Form





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MathML Form







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</ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </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["WeierstrassZeta", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SubscriptBox["\[Eta]", "1"], " ", SuperscriptBox["z", RowBox[List["1", "-", "n"]]]]], RowBox[List[SubscriptBox["\[Omega]", "1"], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "n"]], "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["Csc", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], "]"]], "2"], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]], RowBox[List["4", " ", SubsuperscriptBox["\[Omega]", "1", "2"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "n", " ", SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", SubscriptBox["\[Omega]", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "2"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "z"]], ")"]]]], SubscriptBox["\[Omega]", "1"]]]], "]"]]]], RowBox[List["k", "+", "1"]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[Pi]", RowBox[List["n", "+", "1"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["q", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox["k", "n"]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]", " ", "z"]], SubscriptBox["\[Omega]", "1"]], "+", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"]]], "]"]]]], RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]]]]]]], SubsuperscriptBox["\[Omega]", "1", RowBox[List["n", "+", "1"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29





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