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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 the function itself





http://functions.wolfram.com/01.19.06.0049.01









  


  










Input Form





Sinh[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == I Sum[(1/(2 k)!) (z - (Pi I)/2)^(2 k), {k, 0, n}] == Sinh[z] - ((I Sqrt[Pi])/2^(2 n + 2)) (z - (Pi I)/2)^(2 + 2 n) HypergeometricPFQRegularized[{1}, {3/2 + n, 2 + n}, (1/4) (z - (Pi I)/2)^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", "[", RowBox[List["Sinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List["\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["2", " ", "k"]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]], "\[Equal]", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox["\[Pi]"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]]]], "]"]]]], SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]]]]]]]], "&&", 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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