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
and
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.
