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BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > General case





http://functions.wolfram.com/03.03.06.0036.01









  


  










Input Form





BesselY[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; Subscript[F, m][z, \[Nu]] == Csc[Pi \[Nu]] (Cos[\[Nu] Pi] Sum[((-1)^k (z/2)^(2 k + \[Nu]))/(Gamma[k + \[Nu] + 1] k!), {k, 0, m}] - Sum[((-1)^k (z/2)^(2 k - \[Nu]))/ (Gamma[k - \[Nu] + 1] k!), {k, 0, m}]) == BesselY[\[Nu], z] - ((Csc[Pi \[Nu]] (-1)^m)/(m + 1)!) (((2^(-2 - 2 m + \[Nu]) z^(2 - \[Nu] + 2 m))/Gamma[2 + m - \[Nu]]) HypergeometricPFQ[{1}, {2 + m, 2 + m - \[Nu]}, -(z^2/4)] - ((2^(-2 - 2 m - \[Nu]) z^(2 + \[Nu] + 2 m) Cos[Pi \[Nu]])/ Gamma[2 + m + \[Nu]]) HypergeometricPFQ[{1}, {2 + m, 2 + m + \[Nu]}, -(z^2/4)]) && Element[m, Integers] && m >= 0 /; !Element[\[Nu], Integers]










Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> <apply> <notin /> <ci> &#957; </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Nu]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "m"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Nu]", "+", RowBox[List["2", " ", "m"]]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "-", "\[Nu]"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "m"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]", "+", RowBox[List["2", " ", "m"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "+", "\[Nu]"]], "]"]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]










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





2007-05-02