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 EllipticE

 http://functions.wolfram.com/08.04.20.0005.02

 Input Form

 D[EllipticE[z, m], {z, n}] == KroneckerDelta[n] EllipticE[z, m] + ((2 (2 I)^(n - 1) Sqrt[1 - m Sin[z]^2])/ Sqrt[(2 Sqrt[1 - m] + (1 - E^(2 I z)) m - 2)/Sqrt[1 - m]]) Sqrt[(E^(2 I z) m)/(m - 2 (1 + Sqrt[1 - m]))] Sum[((E^(2 I k z) StirlingS2[n - 1, k] Pochhammer[-(1/2), k] (-m)^k)/ (2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)^k) 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])], {k, 0, 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["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]], SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]]], ")"]], " ", "m"]], "-", "2"]], SqrtBox[RowBox[List["1", "-", "m"]]]]]], " ", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", "m"]], RowBox[List["m", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "z"]]], RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "k"]]], SuperscriptBox[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"]]]], ")"]], "k"]], " ", 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[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n E ( z m ) z n E ( z m ) δ KroneckerDelta n + 2 ( 2 ) n - 1 1 - m sin 2 ( z ) ( 1 - 2 z ) m + 2 1 - m - 2 1 - m 2 z m m - 2 ( 1 - m + 1 ) k = 0 n - 1 2 k z 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] n - 1 ( k ) ( - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( - m ) k ( ( 2 z - 1 ) m + 2 1 - m + 2 ) k 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 EllipticE z m EllipticE z m KroneckerDelta n 2 2 n -1 1 -1 m z 2 1 2 1 -1 2 z m 2 1 -1 m 1 2 -2 1 -1 m 1 2 -1 1 2 -1 2 z m m -1 2 1 -1 m 1 2 1 -1 1 2 k 0 n -1 2 k z StirlingS2 n -1 k Pochhammer -1 1 2 k -1 m k 2 z -1 m 2 1 -1 m 1 2 2 k -1 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 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", "m"]], RowBox[List["m", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", 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"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "k"]]], ")"]], " ", 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"]]]]]]]], "]"]]]], SuperscriptBox[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"]]]], ")"]], "k"]]]]]], SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]]], ")"]], " ", "m"]], "-", "2"]], SqrtBox[RowBox[List["1", "-", "m"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29