Bessel function of the second kind: Integration (formula 03.03.21.0023)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving exponential function and a power function

Involving exp and power

Linear arguments

http://functions.wolfram.com/03.03.21.0023.01

Input Form

Integrate[(z^\[Nu] BesselY[\[Nu], a z])/E^(I a z), z] == (1/(a (1 + 2 \[Nu]) Gamma[-\[Nu]])) ((z^\[Nu] Csc[Pi \[Nu]] ((-I) 2^(1 + \[Nu]) E^(I a z) + I a z (a z)^\[Nu] BesselJ[-1 - \[Nu], a z] Gamma[-\[Nu]] - a z (a z)^\[Nu] BesselJ[-\[Nu], a z] Gamma[-\[Nu]] + a z (a z)^\[Nu] BesselJ[\[Nu], a z] Cos[Pi \[Nu]] Gamma[-\[Nu]] + I a z (a z)^\[Nu] BesselJ[1 + \[Nu], a z] Cos[Pi \[Nu]] Gamma[-\[Nu]]))/ (E^(I a z) (a z)^\[Nu]))

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <imaginaryi /> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> ν </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> ν </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29