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\centerline{Mathematics 174 -- Applied Mathematics 2}
\centerline{Discussion 3/Problem Set 2 -- Bessel Functions}
\centerline{February 10, 2000}
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\noindent
{\it Background}
\bigskip
Recall that last time we began studying the 2D wave equation for 
a circular membrane with its outer edge (at $r = a$) fixed at zero vertical
displacement.  By separation of variables, we were led to the Sturm-Liouville
form equation:
$$ {d\over dr}\left(r {df\over dr}\right) - {m^2\over r}f = -\lambda r f\leqno(1)$$
with the singular boundary conditions $|f(0)| < \infty$ and $f(a) = 0$.
Because $p = r$ and  $\sigma = r$ are not strictly positive on $[0,a]$, and since
$q = - {m^2\over r}$ has a discontinuity at $r = 0$, this {\it is not
a regular Sturm-Liouville problem.}  However, we will see that the 
statements in the Sturm-Liouville theorem {\it are still true here}.

After multiplying by $r$ and making the change of variable 
$z = \sqrt{\lambda} r$, equation (1) reduced
to {\it Bessel's equation of order} $m$:
$$z^2 {d^2f\over dz^2} + z {df\over dz} + (z^2 - m^2)f = 0.$$
We will be most interested in the case that $m$ is a positive integer.
However, Bessel's equation makes sense as a differential equation
for all $m \in {\bf R}$.  
We derived a formal series solution of this equation:
$$J_m(z) = \sum_{\ell = 0}^\infty {(-1)^\ell \over \ell! (m + \ell)!}
\left({z \over 2}\right)^{2\ell + m}\leqno(2)$$
called the {\it Bessel function of the first kind, of order $m$}.
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\noindent
{\it Discussion Questions}
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\item{A)}  Our derivation of the series expansion (2) for $J_m(z)$ was 
completely ``formal'' -- it was based only on the recurrence
relation for the coefficients.  In order to say that the infinite series
in (2) determines
a well-defined function of $z$ on some interval containing zero, {\it what
else do we have to show\/}?  On what interval $(-L,L)$ does (2) specify
a well-defined function of $z$?  
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\item{B)}  The part of the Sturm-Liouville theorem that is easiest
to prove (even without regularity) is the {\it orthogonality relation for eigenfunctions\/}.  
Assume $\lambda_1 \ne \lambda_2$
are distinct eigenvalues of (1) -- that is, assume there exist non-zero $f_1$ and $f_2$
satisfying 
$$\cases{L(f_1) =  
{d\over dr}\left(r {df_1\over dr}\right) - {m^2\over r}f_1 = -\lambda_1 r f_1 &\cr
L(f_2) =  
{d\over dr}\left(r {df_2\over dr}\right) - {m^2\over r}f_2 = -\lambda_2 r f_2 &\cr}$$
and $f_1(a) = f_2(a) = 0$ (and $f_1(0), f_2(0)$ finite).
Deduce that 
$$\langle f_1,f_2 \rangle_r = \int_0^a f_1(r)f_2(r)r\ dr = 0.$$
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\bigskip
\noindent
{\it Problem Set 2}
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\item{C)}  Write out the first 6 non-zero terms in
the expansions of $J_0(z)$ and $J_1(z)$.  Plot the corresponding functions
together on the interval $[0,4]$.
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\item{D)}   It is a general fact that for a function defined by a convergent
power series (as in (2)), the derivative can be found by term-by-term
differentiation. Show that
$${d\over dz}(z^{-m} J_m(z)) = -z^{-m} J_{m+1}(z)$$
for all integers $m \ge 0$ and similarly,
$${d\over dz}(z^m J_m(z)) = z^m J_{m-1}(z)$$
for all integers $m \ge 1$.
If $m = 0$ in particular, this says $J_0'(z) = -J_1(z)$.  Does this
agree with your plots in part C?
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\item{E)}  Prove the following recurrence relation for the Bessel functions
$$J_{n+1}(z) = {2n\over z} J_n(z) - J_{n-1}(z)$$
(Hint: Combine the two relations from question D.)
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\noindent
So far we have concentrated on integer values $m$.  The series expansion
for $J_m(z)$ given above can also be extended to non-integer $m$ if 
we replace the factor $(m + \ell)!$ by a value of an important 
special function called the {\it gamma function}.  The gamma function
is usually defined as follows:  For $x > 0$, 
$$\Gamma(x) = \int_0^\infty e^{-t} t^{x - 1}\ dt$$
It {\it interpolates values of the factorial} in the sense given in part 1
of the following problem.
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\item{F)}  
\itemitem{1)}  Show that $\Gamma(1) = 1$ and $\Gamma(x + 1) = x\Gamma(x)$ (Hint:
Think parts!)
Deduce that for all integers $n \ge 0$, $\Gamma(n + 1) = n!$.
\itemitem{2)}  Explain how the relation $\Gamma(x + 1) = x\Gamma(x)$ can
also be used to define $\Gamma(x)$ for $x < 0$.  Note though:  $\Gamma(0)$
is undefined.  This implies that $\Gamma(n)$ is not defined
either if $n < 0$ is an integer.  Also: the gamma function is
built-in in Maple.  It is called {\tt GAMMA} (all capitals).  Be careful
if you try to plot it on an interval containing a negative integer though.
The gamma function has vertical asymptotes at those points.
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\noindent
Now, we can define 
$$J_m(z) = \sum_{\ell = 0}^\infty {(-1)^\ell \over \ell! \Gamma(m + \ell + 1)}
\left({z \over 2}\right)^{2\ell + m}\leqno(3)$$
and the series in (3) makes sense for non-integer $m$ as well.
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\item{G)}  In this question we will see that 
$$\Gamma\left({1\over 2}\right) = \sqrt{\pi}.\leqno(4)$$
\itemitem{1)} First by definition,
$$\Gamma\left({1\over 2}\right) = \int_0^\infty e^{-t} t^{-1/2} dt$$
Let $t = u^2$, and change variables in the integral.
\itemitem{2)} Next, take two ``copies'' of your integral in part 1, written
with different variables of integration $u,v$.  If you multiply the copies
together, the result can be written as a double integral over the
first quadrant of the $u,v$ plane.  It can be evaluated
by converting to polar coordinates (the polar limits are $r \in [0,\infty)$,
$\theta \in [0,\pi/2]$).  What value do you get?  Deduce (4) above.
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\item{H)}  By calculations similar to those in question G it can 
be shown that for any integer $n \ge 1$, 
$$\Gamma\left(n + {1\over 2}\right) = {1\cdot 3 \cdot 5 \cdots (2n-1)\over 2^n}
\sqrt{\pi}.$$
This implies that
if $m$ is of the form (odd integer)/2, then the Bessel function
$J_m(z)$ can be expressed in terms of ordinary trigonometric functions.
For instance, show that 
$$J_{1/2}(z) = \sqrt{{2\over \pi z}} \sin(z)$$
and 
$$J_{-1/2}(z) = \sqrt{{2\over \pi z}} \cos(z)$$
Find a similar formula for  $J_{3/2}(z)$.
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\noindent
{\it Assignment}
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Group write-ups for questions A,B, and individual write-ups for the others,
due Thursday, February 17.
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