On Bessel Functions

We considered the following second order linear equation

(((d^2)R)/(dx^2)) + (1/x)(dR/dx) + (1-((ν^2)/(x^2)) ) R =0

It is called the Bessel equation, and ν is a parameter that, in general, can be a complex number.  We found a series expression, that converges for all x, for R(x)

R(x) = (Underoverscript[(x^α)∑, (         j=0,2), (        ∞)])(Subscript[a, j]) (x^j)      α= ±ν
(Subscript[a, 2j]) = (Subscript[a, 0])  (((((-1))^j)Γ(1+α))/((2^(2j))j! Γ(j+α+1)))     for j=0,2,4...

The convention is to pick  (Subscript[a, 0])=1/(2^α)Γ(α+1)

Let's construct this series below

a[j_,α_]=  (((-1))^j) /j!/Gamma[j+1+α]

Table[a[j,1/2],{j,0,10}]

Bess[ν_,max_,x_]:=(((x/2))^ν) Sum[ a[j,ν] (((x/2))^(2j)),{j,0,max}]

Plot[Bess[1/2,10,x],{x,0,10}]

Lets compare this with the Mathematica function BesselJ[ν,x]

Plot[{Bess[1/2,10,x],BesselJ[1/2,x]},{x,0,10},PlotStyle->Thick]

How about the Bessel function with the negative index ν

Plot[{Bess[-1/2,10,x],BesselJ[-1/2,x]},{x,0,10},PlotStyle->Thick]

Note that the two function (Subscript[J, ±ν])(x) look very different.  As a matter of fact they are linearly independent.   Let’s 
construct the Wronksian

(((Subscript[dJ, ν])(x))/(dx )) (Subscript[J, -ν])(x) - (((Subscript[dJ, -ν])(x))/(dx )) (Subscript[J, ν])(x)

FullSimplify[D[BesselJ[ν,x],x] BesselJ[-ν,x]-D[BesselJ[-ν,x],x]BesselJ[ν,x]]

So we recognize that it does not vanish except in the case where ν is an integer. What are we supposed to do then?

Define a new linear combination of  (Subscript[J, ±ν])(x). It is called the Neumann function

(Subscript[N, ν])(x) = (((Subscript[J, ν])Cos(νπ)-(Subscript[J, -ν]))/(Sin(πν)))

Let's check its  Wronskian

Neum[ν_,x_]=(BesselJ[ν,x]Cos[ν π]-BesselJ[-ν,x])/Sin[π ν]

FullSimplify[D[BesselJ[ν,x],x] Neum[ν,x]-D[Neum[ν,x],x]BesselJ[ν,x]]

So we see that the Neumann function is always linearly independent of (Subscript[J, ν]). Indeed, Mathematica also has the Neumann
function tabulated, it is called

BesselY[ν,x]

Plot[{BesselY[1,x],BesselJ[1,x]},{x,0,10},PlotStyle->Thick]

Asymptotic behavior

Though we have tabulated numerical values for the Bessel functions readible available, it is still usefull to rely on analytic expressions when available. By looking at the two regions x ⟶0, and x⟶∞ we can replace the Bessel functions with simple analytic expression. From the series expression, given above, we find as x ⟶0

(Subscript[J, ν])(x) ⟶ (1/(Γ(1+ν)))((((x/2)))^ν)

(Subscript[N, ν]) (x) ⟶  -((Γ(ν))/π)((((2/x)))^ν)      If  ν=0   (Subscript[N, ν]) (x) ⟶ (2/π) (  Log((2/x))   +0.5772... )

Japprox[ν_,x_]=1/Gamma[1+ν] (x/2)^ν;

Plot[{Japprox[1/2,x],BesselJ[1/2,x]},{x,0,1}]

Napprox[ν_,x_]=-Gamma[ν]/π (x/2)^(-ν);
Plot[{Napprox[1/2,x],BesselY[1/2,x]},{x,0,1}]

Large x behavior

For large values of the Bessel functions also reduce to simple analytic expression, we find

(Subscript[J, v])(x)  ⟶(Sqrt[ (2/(π x))]) Cos(x - ν (π/2)-(π/4) )

(Subscript[N, v])(x)  ⟶(Sqrt[ (2/(π x))]) Sin(x - ν (π/2)-(π/4) )

Japprox[ν_,x_]=(Sqrt[(2/(π x))]) Cos[x - ν (π/2)-(π/4) ];

Plot[{Japprox[1,x],BesselJ[1,x]},{x,0,10}]

Napprox[ν_,x_]=(Sqrt[(2/(π x))]) Sin[x - ν (π/2)-(π/4) ];

Plot[{Napprox[1,x],BesselY[1,x]},{x,0,10}]

On the zeros of the Bessel functions

Let’s graph a typical Bessel function of order ν

Plot[BesselJ[2,x],{x,0,100}]

Notice that there are many (actually ∞) zero’s of the Bessel function, these are special values of x where

(Subscript[J, ν])(x) =0 

we label them by a subscript e.g. (Subscript[x, 1]) might be the first zero, (Subscript[x, 2]) might be the second zero, and so on...
But wait these zeros also depend on the order ν, since

Plot[{BesselJ[2,x],BesselJ[1/2,x]},{x,0,20},PlotStyle->Thick]

they are typically different. So we use two labels, the convention is

(Subscript[x, νn])

represents the n’th zero of the Bessel function (J) of order ν.  These zero’s are hard to find in general, luckily Mathematica stores tables of them so we have them at our fingertips.

Table[BesselJZero[2,k],{k,1,5}]
N[%]

The zero’s of the Bessel functions paly an important role when we need to fit our solutions to boundary conditions, they are
also important in classifing orthogonality relations for them.

Orthogonality of Bessel functions

In our discussion of the Legendre polynomials, and Fourier fucntions, we dealt with a set of complete functions that are orthogonal, so it is natural to inquire if the same is true for Bessel functions. We will answer that question in the affirmative, however there is a slight catch. We might expect that two Bessel functions of different orders ν,ν’ might be orthogonal. 
That is however not true. Instead we find (see proff on page 115 of the text) that two Bessel functions of the same order, ν
but scaled according to

(Subscript[J, ν])((Subscript[x, νn])(ρ/a) )  is orthogonal (in the region  0 < ρ < a, and with respect to weight factor ρ)  to any  (Subscript[J, ν])((Subscript[x, νm])(ρ/a) )   where n≠m

To understand this we are claiming that

(∫_0^a) ρ (Subscript[J, ν])((Subscript[x, νn])(ρ/a))  (Subscript[J, ν])((Subscript[x, νm])(ρ/a)) dρ =0    for all m≠n. 

Let’s look at an example below. First we pick and order ν, any order ν will do. Then we find the zero’s of that
order, and finally construct (Subscript[J, ν])((Subscript[x, νn])(ρ/a))

order=5;
a=3;
x=Table[N[BesselJZero[order,i]],{i,1,10}]

scaledJ[n_,ρ_]:=BesselJ[order,x[[n]]/a ρ]

set=Table[scaledJ[n,ρ],{n,1,10}]

Plot[set,{ρ,0,a}]

So we construct a set of  (Subscript[J, ν])((Subscript[x, νn])(ρ/a))  n=1,2,...10

set[[1]]
set[[4]]

NIntegrate[set[[1]]set[[8]]ρ,{ρ,0,a}]

We see that these functions are indeed orthogonal, how about when the index n is the same ?
It can be show, in that, case

(∫_0^a) ρ (Subscript[J, ν])((Subscript[x, νn])(ρ/a))  (Subscript[J, ν])((Subscript[x, νn])(ρ/a)) dρ = ((a^2)/2)(Subscript[(J^2), ν+1])((Subscript[ν, νn]))

norm[n_]:=a^2/2 BesselJ[order+1 ,x[[n]]]^2

NIntegrate[set[[1]]set[[1]]ρ,{ρ,0,a}]
norm[1]

From the discussion above,since the set(Subscript[J, ν])((Subscript[x, νn])(ρ/a))is orthogonal we can assume that any function of
ρ,i.e.f(ρ) could be represented by

f(ρ)=(∑_n) (Subscript[A, νn]) (Subscript[J, ν])((Subscript[x, νn])(ρ/a))
where,according to our normalization integral,

(Subscript[A, νn]) = (2/((a^2)(Subscript[(J^2), ν+1])((Subscript[ν, νn])))) (∫_0^a) ρ f(ρ)  (Subscript[J, ν])((Subscript[x, νn])(ρ/a)) dρ

The expansion above for f(ρ) is called the Fourier- Bessel series and will come in usefull in matching boundary conditions in problems with cylinderical symmetry.

Other types of Bessel functions.

In obtaining the Bessel equation we made a certain assumption for the sign of the seperation constant, (k^2),  in Eq. (3.73) in the text. In some problems we need to have the seperation constant take the value  -(k^2) so the Z solutions will be
sinusoidal. In that case we get a different equation for the R(ρ) equation, which can be written as

(((d^2)R)/(dx^2)) + (1/x)(dR/dx) - (1+((ν^2)/(x^2)) ) R =0

Which differs from the Bessel equation above.It is called the modified Bessel equation,however it is closely related to the Bessel equation since if we make an analytic continuation by repacing x⟶i x,and then the modified Bessel function
turns into the Bessel equation.This implies that solutions of the modified   Bessel equation are simply Bessel functions in which the argument x is replaced
by i x.In fact two,common,linear independent solutions to the Modified Bessel equation are

(Subscript[K, ν])(x) and(Subscript[I, ν])(x) defined as

(Subscript[I, ν])(x)≡(i^(-ν)) (Subscript[J, ν])(i x)

(Subscript[K, ν])(x)≡(π/2) (i^(ν+1)) (  (Subscript[J, ν])(i x) + i (Subscript[N, ν])(i x) )

They have asymptotic behavior given in Eqs. 3.102-3.104 in the text.