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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[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/01.19.06.0058.01









  


  










Input Form





Sinh[z]^n \[Proportional] I^n (1 - (1/2^(n + 2)) ((-2^(1 + n)) n + (1 + (-1)^n) Binomial[n, Floor[n/2]] (n - 2 Floor[n/2])^2) (z - (Pi I)/2)^2 + (1/(3 2^(n + 4))) (2^(1 + n) n (-2 + 3 n) - (1 + (-1)^n) Binomial[n, Floor[n/2]] (n - 2 Floor[n/2])^4) (z - (Pi I)/2)^4 + O[(z - (Pi I)/2)^6]) /; 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["Sinh", "[", "z_", "]"]], "n_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["1", "+", "n"]]]]], " ", "n"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]], SuperscriptBox["2", RowBox[List["n", "+", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "n"]]], " ", "n", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "n"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], "4"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "4"]]], RowBox[List["3", " ", SuperscriptBox["2", RowBox[List["n", "+", "4"]]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "]"]], "6"]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02