Homework for Section 5.4
Mathematics 360
due Monday, December 9
Read Elementary Classical Analysis, Section 5.4.
1. Exercises 2–7, pp. 267–268 of ECA.
2. The Bessel function Jis defined by the power series
J(x) =
∞
X
n=0 −1
4nx2n
n!2
Prove that Jis twice differentiable and that Jsatisfies the differential equation
d
dx xdJ
dx +xJ = 0
3. Using the differential equation, show that
d
dx J0(x)2+J(x)2=−2
xJ0(x)2
Use this to prove carefully that both J0and Jare bounded for x≥0. What happens
for x≤0?
4. Using an argument similar to the one used in class to study C(x) (the cosine function),
show that Jhas a positive zero. Hint: first show that xJ0(x) is decreasing if x > 0 and
J(x)>0.
5. Show that Jhas infinitely many positive zeroes. Hint: Suppose we have already found
a zero x1, with J(x1) = 0 and J0(x1)<0. Show that Jis concave up whenever x > 0,
J(x)<0, and J0(x)<0. From this, show that Jhas a local minimum at some point
x > x1. Then use the argument from the previous problem to get the next zero. The
case where J0(x1)>0 is very similar.