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Cosh






Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Series representations > Generalized power series > Expansions at z==Pi i/2 > For powers of the function > For symbolical integer power





http://functions.wolfram.com/01.20.06.0057.01









  


  










Input Form





Cosh[z]^n \[Proportional] I^n (z - (Pi I)/2)^n (1 + ((2^(1 - n)/(2 + n)!) Sum[(-1)^k Binomial[n, k] (n - 2 k)^(n + 2), {k, 0, Floor[(n - 1)/2]}]) (z - (Pi I)/2)^2 + ((2^(1 - n)/(n + 4)!) Sum[(-1)^k Binomial[n, k] (n - 2 k)^(n + 4), {k, 0, Floor[(n - 1)/2]}]) (z - (Pi I)/2)^4 + \[Ellipsis]) /; (z -> (Pi I)/2) && 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", "[", SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "n_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", "2"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]], RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", "4"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "4"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "4"]], ")"]], "!"]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02





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