Bessel Function of the First Kind

Author

Eric W. Weisstein
April 16, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Plots

Real Axis

Plot[Evaluate[Table[BesselJ[n,x],{n,0,5}]],{x,0,10},AxesLabel->{x,BesselJ[n,x]},PlotStyle->{Black,Red,Yellow,Green,Blue,Purple},
PlotLabels->Placed[Automatic,Above]]

Complex Plane

V6

<<Utilities`Plot`
<<Utilities`Typesetting`

AbsReImPlot[BesselJ[0,z],{z,5},ImageSize->Large]

V5

ComplexPlot3D[BesselJ[0,z],{z,-5,5},PlotPoints->100]//Timing

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

Differential Equation

V6

DSolve[x^2 y''[x]+x y'[x]+(x^2-n^2)y[x]==0,y[x],x]

V7

DSolve[x^2 y''[x]+x y'[x]+(x^2-n^2)y[x]==0,y[x],x]

Recurrence Relation

b[l_?OddQ]:=0
b[l_]:=(-1)^(l/2) m!/(l/2)!/2^l/(m+l/2)!

Table[b[n],{n,5}]//InputForm

Factor[FullSimplify[b[n]/b[n-2]]]//InputForm

v4.2

<<DiscreteMath`RSolve`

RSolve[{a[n]==-a[n-2]/n/(2m+n),a[0]==0,a[1]==0},a[n],n]

v5.0

$Version

(soln=RSolve[{a[n]n(2m+n)+a[n-2]==0,a[1](2m+1)==0},a[n],n])//InputForm

Table[a[n]/.soln[[1]],{n,1}]

V7

soln=RSolve[{a[n]n(2m+n)+a[n-2]==0,a[1](2m+1)==0},a[n],n]

FullSimplify[%,(m|n)∈Integers]

Integrals

(∫_0^π)(e^(i z (θ cos)))dθ=(Subscript[J, 0])(z)

Integrate[Exp[I z Cos[θ]],{θ,0,π},Assumptions->Im[z]==0]/π

((2 (z^n) (∫_0^(π/2))(sin^(2 n))(u) cos(z cos(u))du)/(π (2 n-1)!!))==(Subscript[J, n])(z)

(2/π)((z^n)/((2n-1)!!))Integrate[((Sin[u])^(2n))Cos[z Cos[u]],{u,0,π/2},Assumptions->z>0&&n>0]//FullSimplify[#,n∈Integers]&

Sums

(∑_(l=0)^∞)(((((-1))^l) (x^(2 l+m)))/((2^(2 l+m)) l! (l+m)!))=(Subscript[J, m])(x)

V6

Sum[((((-1))^l)/((2^(2l+m))l!(m+l)!))(x^(2l+m)),{l,0,∞}]

V7

Sum[((((-1))^l)/((2^(2l+m))l!(m+l)!))(x^(2l+m)),{l,0,∞}]

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

V6

Sum[(-1)^k (z^2/4)^k/k!^2,{k,0,∞}]

V7

Sum[(-1)^k (z^2/4)^k/k!^2,{k,0,∞}]

(∑_(n=1)^∞)(Subscript[J, n])(z) (t^n)

V6

Sum[BesselJ[n,z]t^n,{n,∞}]

V7

Sum[BesselJ[n,z]t^n,{n,∞}]

(∑_(k=-∞)^∞)(Subscript[J, k])(z)==1

V7

Sum[BesselJ[k,z],{k,-∞,∞}]

V8

Sum[BesselJ[k,z],{k,-∞,∞}]

(∑_(n=-∞)^∞)(t^n) (Subscript[J, n])(z)==(e^((1/2) (t-(1/t)) z))

V6

Sum[BesselJ[n,z]t^n,{n,-∞,∞}]

Sum[Series[BesselJ[n,z],{z,0,50}],{n,-50,50}]

V7

Sum[BesselJ[n,z]t^n,{n,-∞,∞}]

(∑_(n=0)^∞)2 (((Subscript[J, n])(x))^2)-(((Subscript[J, 0])(x))^2)==1

V6

Sum[2BesselJ[n,x]^2,{n,0,Infinity}]-BesselJ[0,x]^2

V7

Sum[2BesselJ[n,x]^2,{n,0,Infinity}]-BesselJ[0,x]^2

(∑_(n=-∞)^∞)(i^n) (e^(i n θ)) (Subscript[J, n])(z)==exp(i z cos(θ))

V6

Sum[I^n BesselJ[n,z]Exp[I n θ],{n,-Infinity,Infinity}]

V7

Sum[I^n BesselJ[n,z]Exp[I n θ],{n,-Infinity,Infinity}]//InputForm

2 (∑_n^∞)(i^n) (Subscript[J, n])(z) cos(n θ)+(Subscript[J, 0])(z)==exp(i z cos(θ))

BesselJ[0,z]+2Sum[(I^n)BesselJ[n,z]Cos[n θ],{n,∞}]//InputForm

(∑_(m=-∞)^∞)(Subscript[J, m])(y) (Subscript[J, n-m])(z)==(Subscript[J, n])(y+z)

Sum[BesselJ[m,y]BesselJ[n-m,z],{m,-∞,∞}]//InputForm

π (∑_(n=0)^∞)(((Subscript[J, n+(1/2)])(x))^2)==Si(2 x)

Pi Sum[BesselJ[n+1/2,x]^2,{n,0,∞}]

Chebyshev Identities

j[n_,x_]:=Module[{l=CoefficientList[ChebyshevT[n,x],x]},I^n Sum[I^(k-1) l[[k]] D[BesselJ[0,x],{x,k-1}],{k,n+1}] ]

i[n_,x_]:=Module[{l=CoefficientList[ChebyshevT[n,x],x]},Sum[l[[k]] D[BesselI[0,x],{x,k-1}],{k,n+1}]]

Simplify[Table[j[n,x],{n,0,5}]]

Simplify[Table[i[n,x],{n,0,5}]]

Nicholson Formula

Nicholson[nu_,z_]:=BesselJ[nu,z]^2+BesselY[nu,z]^2-8/Pi^2 NIntegrate[BesselK[0,2z Sinh[t]]Cosh[2 nu t],{t,0,Infinity},
			Method->Trapezoidal]

Plot[Nicholson[1,z],{z,.2,3},PlotRange->All]

Watson Formula

Watson[mu_,nu_,z_]:=BesselJ[mu,z]BesselY[nu,z]-BesselJ[nu,z]BesselY[mu,z]-4Sin[(mu-nu)Pi]/Pi^2 NIntegrate[BesselK[nu-mu,2z Sinh[t]]Exp[-(mu+nu)t],{t,0,Infinity},Method->Trapezoidal]

WatsonD[nu0_,z_]:=BesselJ[nu,z]D[BesselY[nu,z],nu]-
		BesselY[nu,z]D[BesselJ[nu,z],nu]+4/Pi NIntegrate[BesselK[0,2z Sinh[t]]Exp[-2 nu0 t],{t,0,Infinity},Method->Trapezoidal] /. nu->nu0

Plot[Evaluate[{WatsonD[2,z],WatsonD[3.2,z]}],{z,0,3},PlotRange->All]

Plot[Evaluate[{Watson[2.3,3,z],Watson[2.1,1.5,z]}],{z,.5,3},PlotRange->All]

w={};
Do[
	r=Random[Real,{0,10}];
	AppendTo[w,Watson[r,r-Random[],Random[Real,{.1,10}]]],
	{i,100}
];
w

Weyrich's Formula

Weyrich[x_,k_,r_]:={I/2 NIntegrate[Exp[I τ x]Hankel[1,0,r Sqrt[k^2-τ^2]],{τ,-Infinity,Infinity}],
Exp[I k Sqrt[r^2+x^2]]/Sqrt[r^2+x^2]}

Weyrich[1.2,.3,.4]

Continued Fraction

With[{n=2,z=1.1},{((2n)/z)-((z/(2(n+1)))/(1+ContinuedFractionK[-((((z/2))^2)/((n+k)(n+k+1))),1,{k,20}])),((BesselJ[n-1,z])/(BesselJ[n,z]))}]

TimeConstrained[((2n)/z)-((z/(2(n+1)))/(1+ContinuedFractionK[-((((z/2))^2)/((n+k)(n+k+1))),1,{k,∞}])),60]//Timing