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 EllipticE

 http://functions.wolfram.com/08.04.20.0013.01

 Input Form

 D[EllipticE[z, m], {z, n}] == KroneckerDelta[n] EllipticE[z, m] + Sum[(1/j!) Sum[Binomial[j, q] Sum[((-1)^q 2^(q - j) Sin[z]^q (2 p + q - j)^n Binomial[j - q, p] Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/((j - i - 1)! (2 Sin[z])^(j - 2 i - 1))) Sum[Binomial[i, s] Pochhammer[1/ 2, s] Pochhammer[-(1/2), i - s] m^(i - s) Cos[z]^(-1 - 2 s) (1 - m Sin[z]^2)^(1/2 - i + s), {s, 0, i}], {i, 0, j - 1}])/ E^((1/2) I ((n - 2 p - q + j) Pi + 2 (2 p + q - j) z)), {p, 0, j - q}], {q, 0, j - 1}], {j, 1, n}] /; 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[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[FractionBox["1", RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "q"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "-", "q"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], SuperscriptBox["2", RowBox[List["q", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "q"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "n"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "p"]], "-", "q", "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "z"]]]], ")"]]]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", "q"]], ",", "p"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", "z", "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "i"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["i", ",", "s"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["i", "-", "s"]]]], "]"]], " ", SuperscriptBox["m", RowBox[List["i", "-", "s"]]], SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", "s"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "i", "+", "s"]]]]]]]]], ")"]]]]]]]]]]]]]]]]]]]], "/;", 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 + j = 1 n 1 j ! q = 0 j - 1 ( j q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p = 0 j - q ( - 1 ) q 2 q - j sin q ( z ) ( q + 2 p - j ) n - 1 2 ( π ( j + n - q - 2 p ) + 2 ( - j + q + 2 p ) z ) ( j - q p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "-", "q"]], Identity, Rule[Editable, True]]], List[TagBox["p", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] i = 0 j - 1 ( 1 - j ) 2 ( j - i ) - 2 ( j - i - 1 ) ! ( 2 sin ( z ) ) j - 2 i - 1 TagBox[FractionBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "j"]], ")"]], RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["sin", "(", "z", ")"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]], Pochhammer] s = 0 i ( i s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 2 ) s TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "s"], Pochhammer] ( - 1 2 ) i - s TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["i", "-", "s"]]], Pochhammer] m i - s cos - 2 s - 1 ( z ) ( 1 - m sin 2 ( z ) ) - i + s + 1 2 /; n Condition z n EllipticE z m EllipticE z m KroneckerDelta n j 1 n 1 j -1 q 0 j -1 Binomial j q p 0 j -1 q -1 q 2 q -1 j z q q 2 p -1 j n -1 1 2 j n -1 q -1 2 p 2 -1 j q 2 p z Binomial j -1 q p i 0 j -1 Pochhammer Subscript 1 -1 j 2 j -1 i -2 j -1 i -1 2 z j -1 2 i -1 -1 s 0 i Binomial i s Pochhammer 1 2 s Pochhammer -1 1 2 i -1 s m i -1 s z -2 s -1 1 -1 m z 2 -1 i s 1 2 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"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "q"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "-", "q"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", SuperscriptBox["2", RowBox[List["q", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "q"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "p"]], "-", "q", "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], " ", "z"]]]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", "q"]], ",", "p"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "i"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["i", ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["i", "-", "s"]]]], "]"]], " ", SuperscriptBox["m", RowBox[List["i", "-", "s"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "s"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "i", "+", "s"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", "z", "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]]]]]]]]]]]], RowBox[List["j", "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02