Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on March 12, 2009
due March 17, 2009
(numbering of problems continued from
the last assignment with the same due date)
Problem 4 (from Stein & Shakarchi, p.128, #4). Suppose Qis a polynomial
of degree ≥2 with distinct roots, none lying on the real axis. Calculate
Z∞
−∞
e−2πixξ
Q(x)dx, ξ ∈R
in terms of the roots of Q. What happens when several roots coincide?
Hint: Consider separately the cases ξ < 0, ξ= 0, and ξ > 0. Use residues.
Problem 5 (from Stein & Shakarchi, p.128, #5). More generally, let R(x) =
P(x)
Q(x)be a rational function with (degree Q)≥(degree P) + 2 and Q(x)6= 0
on the real axis.
(a) Prove that if α1,· · · , αkare the roots of Qin the upper half-plane, then
there exist polynomials Pj(ξ) of degree less than the multiplicity of αjso
that
Z∞
−∞
R(x)e−2πixξ dx =
k
X
j=1
Pj(ξ)e−2πiαjξ,when ξ < 0.
(b) In particular, if Q(x) has no zeros in the upper half-plane, then
Z∞
−∞
R(x)e−2πixξ dx = 0
for ξ < 0.
(c) Show that similar results hold in the case ξ > 0.
(d) Show that
Z∞
−∞
R(x)e−2πixξ dx =O¡e−a|ξ|¢, ξ ∈R
as |ξ| → ∞ for some a > 0. Determine the best possible a’s in terms of the
roots of R.
