Bessel Function of the Second Kind

Author

Eric W. Weisstein
August 19, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/BesselFunctionoftheSecondKind.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`

Real Axis

Plot[Evaluate[Table[BesselY[n,x],{n,0,5}]],{x,0,10},PlotRange->{-1,.55},AxesLabel->TraditionalForm/@{x,BesselY[n,x]},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet}]

Complex Plane

ComplexPlot3D[BesselY[0,z],{z,-5,5}]//Timing

ComplexContourPlot[BesselY[0,z],{z,-5,5}]//Timing

Sums

((2 ln((z/2)))/π)(Subscript[J, n])(z)-(1/π)((((z/2)))^(-n)) (∑_(k=0)^(n-1))(((((z^2)/4)))^k)(( (-k+n-1)!)/(k!))-(1/π)((((z/2)))^n) (∑_(k=0)^∞)(((-((z^2)/4)))^k)(( ((ψ^((0)))(k+n+1)+(ψ^((0)))(k+1)))/(k! (k+n)!))

SeriesY[n_,z_]:=-(1/2z)^(-n)/Pi Sum[(n-k-1)!/k!(z^2/4)^k,{k,0,n-1}]+2/Pi Log[z/2]BesselJ[n,z]-(z/2)^n/Pi Sum[(PolyGamma[0,k+1]+PolyGamma[0,n+k+1])(-z^2/4)^k/k!/(n+k)!,{k,0,∞}]

(Subscript[Y, 0])(z)

V6

SeriesY[0,z]

FullSimplify[SeriesY[0,z]]

Simplify[Series[BesselY[0,z],{z,0,3}]]

Plot[{((2 (EulerGamma+Log[(z/2)]))/π),BesselY[0,z]},{z,0,5},
PlotStyle->{Blue,Red},
AxesLabel->TraditionalForm/@{x,BesselY[0,z]}]

Plot[Log[(z/2)] (z^2),{z,0,1}]

V7

SeriesY[0,z]

FullSimplify[%]

(Subscript[Y, 1])(z)

V6

Series[BesselY[1,z],{z,0,5}]

FullSimplify[SeriesY[1,z]]

FullSimplify[SeriesY[1,z],z>0]

FullSimplify[SeriesY[1,z],z==0]

FullSimplify[SeriesY[1,z],z!=0]

FullSimplify[SeriesY[1,z],z<0]

V7

SeriesY[1,z]

FullSimplify[%]

FullSimplify[%-BesselY[1,z]]

Table[(1/(π (z^2)))(-z+(Sqrt[(z^2)])) (4-2 BesselJ[0,z]+π z BesselY[1,z]-2 z BesselJ[1,z] (EulerGamma+Log[(z/2)]))/.z->RandomComplex[],{100}]//Chop//Union

(Subscript[Y, 2])(z)

V7

SeriesY[2,z]

FullSimplify[%]

(Subscript[Y, n])(z)

V6

Series[BesselY[n,z],{z,0,3}]

V7

-(1/2z)^(-n)/Pi Sum[(n-k-1)!/k!(z^2/4)^k,{k,0,n-1}]+2/Pi Log[z/2]BesselJ[n,z]-(z/2)^n/Pi Sum[(PolyGamma[0,k+1]+PolyGamma[0,n+k+1])(-z^2/4)^k/k!/(n+k)!,{k,0,∞}]

Sum[(n-k-1)!/k!(z^2/4)^k,{k,0,n-1}]

TimeConstrained[Sum[(PolyGamma[0,k+1]+PolyGamma[0,n+k+1])(-z^2/4)^k/k!/(n+k)!,{k,0,∞}],60]//Timing

(2/π) ((∑_k^∞)((((-1))^(k+1))/(((k!))^2)) (Subscript[H, k]) (((((z^2)/4)))^k)+(ln((z/2))+ℽ) (Subscript[J, 0])(z))

v4.2

y=FullSimplify[2/Pi((Log[z/2]+EulerGamma)BesselJ[0,z]+Sum[(z^2/4)^k (-1)^(k+1)HarmonicNumber[k]/k!^2,{k,∞}])]

FullSimplify[y]

Limit[y,z->0]//Timing

FullSimplify[y,z≠0&&z∈Reals]

FullSimplify[y,z>0]

FullSimplify[y,z<0]

v5.0

y=FullSimplify[2/Pi((Log[z/2]+EulerGamma)BesselJ[0,z]+Sum[(z^2/4)^k (-1)^(k+1)HarmonicNumber[k]/k!^2,{k,∞}])]

Limit[y,z->0]//Timing

V7

2/Pi((Log[z/2]+EulerGamma)BesselJ[0,z]+Sum[(z^2/4)^k (-1)^(k+1)HarmonicNumber[k]/k!^2,{k,∞}])

Integrals

Integrate[Sin[z Sin[θ]-n θ],{θ,0,π}]/Pi-Integrate[(Exp[n t]+Exp[-n t](-1)^n)Exp[-z Sinh[t]],{t,0,∞}]/Pi

-2(x/2)^(-ν)/Sqrt[π]/Gamma[1/2-ν] Integrate[Cos[x t]/(t^2-1)^(ν+1/2),{t,1,∞},Assumptions->{-1/2<Re[ν]<1/2,x>0}]