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 JacobiZeta

 http://functions.wolfram.com/08.07.20.0008.01

 Input Form

 D[JacobiZeta[z, m], {z, n}] == KroneckerDelta[n] JacobiZeta[z, m] + KroneckerDelta[n - 1] (Sqrt[1 - m Sin[z]^2] - (EllipticE[m]/EllipticK[m]) (1/Sqrt[1 - m Sin[z]^2])) - ((2 I^(n - 1) Pochhammer[-(1/2), n])/ (n - 1)!) Sum[(((-1)^q Binomial[n - 1, q])/(1 - m Sin[z]^2)^q) (Sqrt[1 - m Sin[z]^2]/(1 - 2 q) + (EllipticE[m]/EllipticK[m]) ((1 - 2 n)/((1 + 2 q) Sqrt[1 - m Sin[z]^2]))) Sum[Binomial[q, j] m^j (2 - m)^(q - j) 2^(n - j - q - 1) Sum[Binomial[j, i] (2 i - j)^(n - 1) E^(2 (2 i - j) I z), {i, 0, j}], {j, 0, q}], {q, 1, n - 1}] /; 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["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " "]], RowBox[List["EllipticK", "[", "m", "]"]]], FractionBox["1", RowBox[List[" ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["2", SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "1"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "n"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "q"]], "]"]]]], RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], "q"]]]], RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], RowBox[List["1", "-", RowBox[List["2", " ", "q"]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " "]], RowBox[List["EllipticK", "[", "m", "]"]]], FractionBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", "n"]]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "q"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], " ", SuperscriptBox["m", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]], RowBox[List["q", "-", "j"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", "j", "-", "q", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], " ", "\[ImaginaryI]", " ", "z"]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Ζ ( z m ) z n δ KroneckerDelta n Ζ ( z m ) + δ KroneckerDelta n - 1 ( 1 - m sin 2 ( z ) - E ( m ) K ( m ) 1 - m sin 2 ( z ) ) - 2 n - 1 ( - 1 2 ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "n"], Pochhammer] ( n - 1 ) ! q = 1 n - 1 ( - 1 ) q ( n - 1 q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m sin 2 ( z ) ) q ( E ( m ) ( 1 - 2 n ) K ( m ) ( ( 2 q + 1 ) 1 - m sin 2 ( z ) ) + 1 - m sin 2 ( z ) 1 - 2 q ) j = 0 q ( q j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["q", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] m j ( 2 - m ) q - j 2 - j + n - q - 1 i = 0 j ( j i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 i - j ) n - 1 2 ( 2 i - j ) z /; n Condition z n JacobiZeta z m KroneckerDelta n JacobiZeta z m KroneckerDelta n -1 1 -1 m z 2 1 2 -1 EllipticE m EllipticK m 1 -1 m z 2 1 2 -1 -1 2 n -1 Pochhammer -1 1 2 n n -1 -1 q 1 n -1 -1 q Binomial n -1 q 1 -1 m z 2 q -1 EllipticE m 1 -1 2 n EllipticK m 2 q 1 1 -1 m z 2 1 2 -1 1 -1 m z 2 1 2 1 -1 2 q -1 j 0 q Binomial q j m j 2 -1 m q -1 j 2 -1 j n -1 q -1 i 0 j Binomial j i 2 i -1 j n -1 2 2 i -1 j z 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["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], "-", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "1"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "n"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "q"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], RowBox[List["1", "-", RowBox[List["2", " ", "q"]]]]], "+", FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]], ")"]]]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "q"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]], ")"]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], " ", SuperscriptBox["m", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]], RowBox[List["q", "-", "j"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", "j", "-", "q", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], " ", "\[ImaginaryI]", " ", "z"]]]]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], "q"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02