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 Cosh

 http://functions.wolfram.com/01.20.20.0014.01

 Input Form

 D[Cosh[z]^m, {z, n}] == Sum[Binomial[m, k] (2 k - m)^n ((1 + (-1)^n) Sum[Binomial[m - 2 k, 2 j] Sinh[z]^(2 j) Cosh[z]^(m - 2 k - 2 j), {j, 0, Floor[m/2] - k}] - (1 - (-1)^n) Sum[Binomial[m - 2 k, 2 j + 1] Sinh[z]^(2 j + 1) Cosh[z]^(m - 2 k - 2 j - 1), {j, 0, Floor[(m - 1)/2] - k}]), {k, 0, Floor[(m - 1)/2]}]/2^m /; Element[m, Integers] && m >= 0 && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "m"]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], "n"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["m", "2"], "]"]], "-", "k"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", "k"]]]], ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], RowBox[List["2", " ", "j"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", "k"]], "-", RowBox[List["2", " ", "j"]]]]]]]]]]], " ", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]], "-", "k"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", "k"]]]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", "k"]], "-", RowBox[List["2", " ", "j"]], "-", "1"]]]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 n cosh m ( z ) z n 2 - m k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 k - m ) n ( ( 1 + ( - 1 ) n ) j = 0 m 2 - k ( m - 2 k 2 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["2", " ", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sinh 2 j ( z ) cosh - 2 j - 2 k + m ( z ) - ( 1 - ( - 1 ) n ) j = 0 m - 1 2 - k ( m - 2 k 2 j + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sinh 2 j + 1 ( z ) cosh - 2 j - 2 k + m - 1 ( z ) ) /; m n + Condition z n z m 2 -1 m k 0 m -1 2 -1 Binomial m k 2 k -1 m n 1 -1 n j 0 m 2 -1 -1 k Binomial m -1 2 k 2 j z 2 j z -2 j -1 2 k m -1 1 -1 -1 n j 0 m -1 2 -1 -1 k Binomial m -1 2 k 2 j 1 z 2 j 1 z -2 j -1 2 k m -1 m n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "m_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["m", "2"], "]"]], "-", "k"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], RowBox[List["2", " ", "j"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", " ", "k"]], "-", RowBox[List["2", " ", "j"]]]]]]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]], "-", "k"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", " ", "k"]], "-", RowBox[List["2", " ", "j"]], "-", "1"]]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02

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