MATH 675 – HOMEWORK #1
DUE 31 JANUARY 2007
Exercise 1. Given Euler’s formula eix = cos(x) + isin(x) derive the trig identities
2 sin(θ) sin(φ) = cos(θ−φ)−cos(θ+φ),
2 sin(θ) cos(φ) = sin(θ−φ) + sin(θ+φ).
Exercise 2. Exercise 5, page 26 in Stein and Shakarchi.
Exercise 3. (a) Show that if Fand Gare twice differentiable functions then the function
u(x, t) = F(x+t) + G(x−t) (1)
is a solution to the wave equation utt =uxx. This solution is known as the travelling wave
solution to the wave equation.
(b) Show that the standing wave solution derived in class, namely
u(x, t) =
∞
X
m=1
(amcos(mt) + bmsin(mt)) sin(mx)
f(x) = u(x, 0) =
∞
X
m=1
amsin(mx)g(x) = ut(x, 0) =
∞
X
m=1
mbmsin(mx)
satisfies (1) with
F(x) + G(x) = f(x) and F0(x)−G0(x) = g(x).
(Hint: For these calculations you may manipulate all infinite series formally without worrying
about convergence.)
Exercise 4. Exercise 2, page 59, Stein and Shakarchi.
