PHZ 3113 Fall 2010
Homework #9, Due Monday, December 6
1. Let the differential operator
d
d
(1 − x2 )
dx
dx
act on functions defined on the interval (−1, 1).
D=
(a) Suppose DPm (x) = −m(m + 1) Pm (x) and DPn (x) = −n(n + 1) Pn (x) Show that for
n 6= m Pm (x) and Pn (x) are orthogonal on (−1, 1).
(b) Verify that P0 = 1 and P1 = x are solutions to DPm = −m(m + 1) Pm . What are the
corresponding values of m?
(c) Let P2 = c0 + c1 x + c2 x2 be a polynomial of degree 2. Choose c0 and c1 so that P2 is
orthogonal to P0 and P1 . Choose c2 so that P2 (1) = 1. Write your resulting P2 (x). Is your
P2 a solution of DP2 = −m(m + 1) P2 for some m? If so, what is the your value for m?
2. Let
f (x) =
sin x 0 < x < π ,
0
π < x < 2π .
(a) Compute the Fourier series coefficients an , bn .
(b) Compute the value of the sum
(c) Compute the value of the sum
∞
X
(−1)k+1
.
(2k)2 − 1
k=1
∞
X
k=1
3. Let
f (x) =
1
[(2k)2 − 1]2
[Hint: what is f ( π2 )?]
.
cos(k0 x) −a < x < a,
0
otherwise.
with k0 a = π2 .
(a) Compute the Fourier transform f̃ (k). What happens at k → k0 ?
(b) Compute the integral (there is an easy way and an easier way to do this)
Z ∞
cos y dy
.
π 2
2
−∞ [( 2 ) − y ]
(c) Compute the integral
Z ∞
cos2 y dy
.
π 2
22
−∞ [( 2 ) − y ]