Here I'll go through a more physical way of viewing Bessel functions. Bessel functions occur often in the study of problems with cylindrical symmetry. So when you see cylindrical symmetry think ``Bessel functions", spherical symmetry think ``Legendre Polynomials" and when you see Cartesians think ``sine and cosine".
Suppose we want to solve in cylindrical coordinates. Write . This substitution, a la separation of variables, leads to the equations
The in the last equation is just 2 dimensional, different from the original used above which is three dimensional. Eqn. (2) is often refered to as Helmoltz's equation. To solve it we could use two methods. The first is to seperate variables into polar coordinates . This gives
which has solutions where n is an integer. This is the same as Boas chapter 13 equation (5.6). The equation for R, Boas(5.7) is
We'd like to know how to solve this equation, which is closely related to Bessel's equation. We don't know how to solve it so we have two choices. One is to do a power series expansion as is done in chapter 12 of Boas. Instead we can backtrack to eqn. (2) and solve it in Cartesian coordinates. Doing separation of variables again with , we obtain
with as a condition on the two constants and that is obtained when you go through separation of variables. So F(x,y) can be written in a rather nice form:
So the general solution can be written
The physical interpretation of this is as follows. is a plane wave travelling in the direction. Its magnitude is restricted to be K. So the general solution to eqn. (2) is the sum of plane waves all with the same wavelength (or wave-vector), travelling in any arbitrary direction. The coefficient indicates the amplitude and phase of a wave travelling in the direction of .
Since there are a continuous range of angles that the wave could go in, we should actually write eqn. (7) as an integral over all possible angles. So writing and we can rewrite eqn. (7) as
Here is the angle the is pointing relative to the x-axis. Letting and noticing that the integrand is periodic, we can rewrite this as
This is the general solution to the two dimensional Helmoltz equation.
Now how do we relate this to eqn. (4) above? This was obtained by saying we wanted a special solution that looked like
So we look for solutions to eqn. (9) which are of this form. That is we have to hunt for the appropriate . Its not impossible to see that does the trick! This gives
Well this does indeed seem to have separated out the r and components into the desired form. So comparing with eqn. (10) we see that
This integral will be defined to be equal to a special function. We'll call it . The is just a pesty normalization factor that we must include but is quite uninteresting. The big news is the thing . This is called a ``Bessel function of the first kind and order n''. The above integral is an integral representation of that function. And this by construction is a solution to eqn. (4). There is a closely related form to the above integral. Let . Then
By noting that we have
In summary, Bessel functions can be thought of as the sum of two dimensional plane waves going in all possible directions.