Bessel function of the second kind: Series representations (formula 03.03.06.0010)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Series representations

Asymptotic series expansions

Expansions for any z in exponential form

Using exponential function with branch cut-containing arguments

http://functions.wolfram.com/03.03.06.0010.01

Input Form

BesselY[\[Nu], z] \[Proportional] (Csc[Pi \[Nu]]/(Sqrt[2 Pi] (z^2)^(1/4))) (E^(-((I Pi)/4) + I Sqrt[z^2]) ((z^\[Nu] Cos[Pi \[Nu]])/ (E^((Pi I \[Nu])/2) (z^2)^(\[Nu]/2)) - (E^((I Pi \[Nu])/2) (z^2)^(\[Nu]/2))/z^\[Nu]) HypergeometricPFQ[{1/2 + \[Nu], 1/2 - \[Nu]}, {}, -(I/(2 Sqrt[z^2]))] + E^((I Pi)/4 - I Sqrt[z^2]) ((E^((I Pi \[Nu])/2) z^\[Nu] Cos[Pi \[Nu]])/ (z^2)^(\[Nu]/2) - (z^2)^(\[Nu]/2)/(E^((Pi I \[Nu])/2) z^\[Nu])) HypergeometricPFQ[{1/2 + \[Nu], 1/2 - \[Nu]}, {}, I/(2 Sqrt[z^2])]) /; (Abs[z] -> Infinity) && !Element[\[Nu], Integers]

Standard Form

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MathML Form

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type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times 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<power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </list> <list /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <notin /> <ci> ν </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29