PHYSICS 116C
Homework 2
Due in class, Thursday October 10.
There are questions on both sides of the sheet.
Office Hours
My office hour: Wednesdays, 10:00–12:00, ISB 212
TA’s office hour: Mondays, 12:00–2:00 pm., ISB 314
Solutions
Typed up homework solutions will be posted after the homework has been handed in. The solutions
are password protected. The userid is 116C and the password will be given in class.
Exams
The midterm will be in class, Tuesday November 5.
The final exam will be on Tuesday December 10, 4:00–7:00 pm.
Exam policy
• No calculators or other electronic devices.
• Closed book. You can bring one sheet of notes that you have prepared your self.
• If you are ill and so cannot take take an exam, it is important that you let me know by phone,
(831) 459-4151, or email, petery@ucsc.edu, in advance. Only in exceptional circumstances will I
accept a “no-show” due to illness that I was not informed of in advance.
Now for the questions.
1. Show that
Z
1
−1
Plm (x)Pkm (x) dx = 0,
if l 6= k ,
where the Plm (x) are associated Legendre polynomials. In other words, associated Legendre
polynomials with different l but the same m are orthogonal in the interval −1 to 1.
Hint: Use the differential equation satisfied by the associated Legendre polynomials and the
method we used in class to show that the Legendre polynomials are orthogonal.
2. Use the ratio test to show that the series for Jν (x) converges for all x.
3. From the series for Bessel functions show that
d
[xJ1 (x)] = xJ0 (x) .
dx
Note: This is a special case of a more general result (d/dx)[xν Jν (x)] = xν Jν−1 (x).
4. From the series expansion show that
J1/2 (x) =
1
r
2
sin(x) .
πx
5. (a) Consider the generating function for Bessel functions
∞
X
e(x/2)(t−1/t) =
Jn (x)tn
n=−∞
and make the substitution t = eiθ to show that
eix sin θ = cos(x sin θ) + i sin(x sin θ)
= J0 (x) + 2 [J2 (x) cos 2θ + J4 (x) cos 4θ + · · ·] + 2i [J1 (x) sin θ + J3 (x) sin 3θ + · · ·] .
(b) Noting that this is a Fourier series, and recalling the orthogonality of sines and cosines in
the interval 0 to π, i.e.
Z
π
cos nθ cos mθ dθ =
Z0 π
Z 0π
sin nθ sin mθ dθ =
π
δnm
2
π
δnm
2
cos nθ sin mθ dθ = 0 ,
0
in which n and m are positive integers, show that
Jn (x) =

Z
1 π


cos(x sin θ) cos nθ dθ ,


 π 0
Z


1



π
(c) Hence show that
1
Jn (x) =
π
Z
n even,
π
sin(x sin θ) sin nθ dθ ,
n odd.
0
π
0
cos(nθ − x sin θ) dθ ,
n = 0, 1, 2, · · · .
and hence, as a special case, one has
1
J0 (x) =
π
Z
π
cos(x sin θ) dθ .
0
(d) From the last result, using a change of variables, show that
2
J0 (x) =
π
Z
1
0
cos xt
√
dt ,
1 − t2
which is a cosine Fourier transform.
Note: This question illustrates that Bessel functions appear when doing various integrals, as well
as in the solution of Bessel’s equation.
6. Find the solution of the following equation in terms of Bessel functions:
y ′′ + 4x2 y = 0 .
Hint: Express this equation in the form of Eq. (16.1) in Ch. 12 of Boas, for suitable choices of the
parameters.
2