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 JacobiZeta

 http://functions.wolfram.com/08.07.20.0005.01

 Input Form

 D[JacobiZeta[z, m], {z, n}] == ((-(((2 I)^(n - 1) Sqrt[Pi])/Sqrt[1 - m Sin[z]^2])) Sum[(((E^(2 I k z) StirlingS2[n - 1, k])/Gamma[3/2 - k]) ((EllipticE[m]/EllipticK[m]) (2 k - 1) m (2 - 2 Sqrt[1 - m] + (E^(2 I z) - 1) m) AppellF1[1/2, -(1/2), 1/2, 1/2 - k, (2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)/(2 + 2 Sqrt[1 - m] - m), (2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)/(4 Sqrt[1 - m])] - (m - 2 E^(2 I z) (m - 2) + E^(4 I z) m) (2 - 2 Sqrt[1 - m] + (Sqrt[1 - m] - 2) m) AppellF1[3/2, 1/2, -(1/2), 3/2 - k, (2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)/(2 + 2 Sqrt[1 - m] - m), (2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)/(4 Sqrt[1 - m])]))/ (((E^(2 I z) - 1) m + 2 Sqrt[1 - m] + 2)/m)^k, {k, 0, n - 1}])/ (((E^(2 I z) (m + 2 Sqrt[1 - m] - 2))/m)^2^(-1) ((m[1 - E^(2 I z)] + 2 Sqrt[1 - m] - 2)/Sqrt[1 - m])^2^(-1)) /; Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 n Ζ ( z m ) z n - ( 2 ) n - 1 π 4 1 - m m 1 - m sin 2 ( z ) ( 2 z ( m + 2 1 - m - 2 ) m ) - 1 / 2 ( m ( 1 - 2 z ) + 2 1 - m - 2 1 - m ) - 1 / 2 k = 0 n - 1 2 k z Γ ( 3 2 - k ) ( ( 2 z - 1 ) m + 2 1 - m + 2 m ) - k 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] n - 1 ( k ) ( E ( m ) ( 2 k - 1 ) m K ( m ) ( ( 2 z - 1 ) m - 2 1 - m + 2 ) F 1 AppellF1 ( 1 2 ; - 1 2 , 1 2 ; 1 2 - k ; ( 2 z - 1 ) m + 2 1 - m + 2 2 1 - m + 2 - m , ( 2 z - 1 ) m + 2 1 - m + 2 4 1 - m ) - ( m - 2 2 z ( m - 2 ) + 4 z m ) ( ( 1 - m - 2 ) m - 2 1 - m + 2 ) F 1 AppellF1 ( 3 2 ; 1 2 , - 1 2 ; 3 2 - k ; ( 2 z - 1 ) m + 2 1 - m + 2 - m + 2 1 - m + 2 , ( 2 z - 1 ) m + 2 1 - m + 2 4 1 - m ) ) /; n + Condition z n JacobiZeta z m -1 2 n -1 1 2 4 1 -1 m 1 2 m 1 -1 m z 2 1 2 -1 2 z m 2 1 -1 m 1 2 -2 m -1 -1 2 m 1 -1 2 z 2 1 -1 m 1 2 -2 1 -1 m 1 2 -1 -1 2 k 0 n -1 2 k z Gamma 3 2 -1 k -1 2 z -1 m 2 1 -1 m 1 2 2 m -1 -1 k StirlingS2 n -1 k EllipticE m 2 k -1 m EllipticK m -1 2 z -1 m -1 2 1 -1 m 1 2 2 AppellF1 1 2 -1 1 2 1 2 1 2 -1 k 2 z -1 m 2 1 -1 m 1 2 2 2 1 -1 m 1 2 2 -1 m -1 2 z -1 m 2 1 -1 m 1 2 2 4 1 -1 m 1 2 -1 -1 m -1 2 2 z m -2 4 z m 1 -1 m 1 2 -2 m -1 2 1 -1 m 1 2 2 AppellF1 3 2 1 2 -1 1 2 3 2 -1 k 2 z -1 m 2 1 -1 m 1 2 2 -1 m 2 1 -1 m 1 2 2 -1 2 z -1 m 2 1 -1 m 1 2 2 4 1 -1 m 1 2 -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["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", FractionBox["1", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List["m", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "2"]], ")"]]]], "m"]]], " ", FractionBox["1", SqrtBox[FractionBox[RowBox[List[RowBox[List["m", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]]], "]"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "2"]], SqrtBox[RowBox[List["1", "-", "m"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "z"]]], " ", RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", "2"]], "m"], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], " ", "m", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "m"]]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]]]], "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "\[ImaginaryI]", " ", "z"]]], " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], "-", "2"]], ")"]], " ", "m"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", "k"]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "m"]]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "k"]], "]"]]]]]]], SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]], "/;", 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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