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 WeierstrassZeta

 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

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

 MathML Form

 n ζ ( z ; g 2 , g 3 ) TagBox[RowBox[List["\[Zeta]", "(", 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[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]] z n - n 2 ( π ω 1 ) n + 1 ( k = 0 n - 1 j = 0 k - 1 ( - 1 ) j 2 - 2 k k + 1 ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sin - 2 k - 2 ( π z 2 ω 1 ) ( 2 k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( k - j ) n - 1 sin ( π z ( k - j ) ω 1 + n π 2 ) ) - π 2 δ KroneckerDelta n - 1 4 ω 1 2 csc 2 ( π z 2 ω 1 ) + z 1 - n η 1 ω 1 Γ ( 2 - n ) + 2 π n + 1 ω 1 n + 1 k = 1 q 2 k k n 1 - q 2 k sin ( π n 2 + k π z ω 1 ) /; n + Condition z n WeierstrassZeta z Subscript g 2 Subscript g 3 -1 n 2 -1 Subscript ω 1 -1 n 1 j 0 k -1 k 0 n -1 -1 j 2 -2 k k 1 -1 Binomial n -1 k z 2 Subscript ω 1 -1 -2 k -2 Binomial 2 k j k -1 j n -1 z k -1 j Subscript ω 1 -1 n 2 -1 -1 2 KroneckerDelta n -1 4 Subscript ω 1 2 -1 z 2 Subscript ω 1 -1 2 z 1 -1 n Subscript η 1 Subscript ω 1 Gamma 2 -1 n -1 2 n 1 Subscript ω 1 n 1 -1 k 1 q 2 k k n 1 -1 q 2 k -1 n 2 -1 k z Subscript ω 1 -1 n SuperPlus [/itex]

 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