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BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/03.03.06.0026.01









  


  










Input Form





BesselY[\[Nu], z] == Sum[(1/(Subscript[z, 0]^k k!)) Sum[(-1)^(m + k) Binomial[k, m] Pochhammer[-\[Nu], k - m] Sum[(((-1)^(i - 1) 2^(2 i - m) Pochhammer[-m, 2 (m - i)] Pochhammer[\[Nu], i])/(m - i)!) ((Subscript[z, 0]/2) Sum[((i - j - 1)!/(j! (i - 2 j - 1)! Pochhammer[1 - i - \[Nu], j] Pochhammer[\[Nu], j + 1])) (Subscript[z, 0]^2/4)^j BesselY[\[Nu] - 1, Subscript[z, 0]], {j, 0, i - 1}] - Sum[((i - j)!/(j! (i - 2 j)! Pochhammer[1 - i - \[Nu], j] Pochhammer[\[Nu], j])) (Subscript[z, 0]^2/4)^j BesselY[\[Nu], Subscript[z, 0]], {j, 0, i}]) (z - Subscript[z, 0])^k, {i, 0, m}], {m, 0, k}], {k, 0, Infinity}] /; Abs[Arg[Subscript[z, 0]]] < Pi










Standard Form





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MathML Form







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</ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> <apply> <ci> BesselY </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> i </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SubsuperscriptBox["zz", "0", RowBox[List["-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "i"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "i"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SubscriptBox["zz", "0"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["i", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubsuperscriptBox["zz", "0", "2"], "4"], ")"]], "j"], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "i"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubsuperscriptBox["zz", "0", "2"], "4"], ")"]], "j"], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "j"]], "]"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]], "!"]]]]]]]]]]], RowBox[List["k", "!"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", SubscriptBox["zz", "0"], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.