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Sech






Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Differentiation > Symbolic differentiation





http://functions.wolfram.com/01.24.20.0006.01









  


  










Input Form





D[Sech[z], {z, n}] == (-1)^n I Sum[(-(1/2))^k k! StirlingS2[n, k] ((1 + Tanh[(1/4) ((-I) Pi + 2 z)]) (1 - Tanh[(1/4) ((-I) Pi + 2 z)])^k - 2^n (1 + Tanh[z]) (1 - Tanh[z])^k), {k, 0, n}] /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["Sech", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], " ", RowBox[List["k", "!"]], " ", RowBox[List["StirlingS2", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Tanh", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["Tanh", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]], "]"]]]], ")"]], "k"]]], "-", RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Tanh", "[", "z", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["Tanh", "[", "z", "]"]]]], ")"]], "k"]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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