Supplementary Problems for Chapter 5
1. Recall (previous problem set) Bessel’s equation:
z 2 w00 + zw0 + (z 2 − n2 )w = 0
(1)
with indicial equation µ2 − n2 = 0.
(a) Assume that n is a positive integer. Show that an attempt to find
a series solution with the negative index fails.
(b) Assume that n is half an odd integer, i.e., n = m + 1/2 where m is
an integer. Find the series solution for the positive index. Show
that an attempt to find a series solution for the negative index
does not fail.
2. Let {u1 (z)}ni=1 be analytic in a domain D of the complex plane and
suppose their Wronskian determinant vanishes identically there. Show
that they are linearly dependent on D (you may assume the statement
is true for n = 2 and proceed by induction).
3. Consider a system w0 = A(z)w where A is analytic in the disk centered
at the origin with radius r, but has an isolated singularity at z0 with
|z0 | = r where at least one entry aij (z) is unbounded. Show that if a
fundamental matrix solution Φ(z) remains analytic in a neighborhood
of z0 , its determinant must vanish at z0 .
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