1. Homework 8
(1) In this exercise, you are going to prove the fundamental Theorem of algebra: any nonconstant complex polynomial must have a root in C. Let us prove it by contradic- tion. Let p(z) = anzn+· · ·+a1z+a0 be a complex polynomial. Assume that p(z) 6= 0 for all z ∈ C. Define a function f : [0, ∞) × [0, 2π] → R by f (r, θ) = 1/p(reiθ). Define I : [0, ∞) → C by
I(r) = Z 2π
0
f (r, θ)dθ.
(a) Prove that fθ = irfr and that I0(r) = 0.
(b) Prove that lim
r→∞I(r) = 0. (Hint: show that when r is sufficiently large, |p(reiθ)| >
|an|rn/2.) (c) Prove that lim
r→0+I(r) = I(0). Find I(0). (Hint: use the continuity of f (r, θ) to estimate |I(r) − I(0)| when r > 0.)
(d) Use the above results to show that such p does not exist.
(2) Let U be an open subset of R2 and [a, b] × [c, d] be a compact subinterval of U.
Suppose f : U → R is C1. Prove that d
dt Z t
a
f (x, t)dx = Z t
a
ft(x, t)dx + f (t, t).
(3) For t > 0, we define
F (t) = Z t
0
log(1 + tx) 1 + x2 dx.
(a) Compute F0(t).
(b) Prove that F (t) = 1
2log(1 + t2) tan−1t for t > 0.
(c) Evaluate Z 1
0
log(1 + x) 1 + x2 dx.
(4) Prove that
Z ∞ 0
(e−x− e−tx)dx
x = log t
for t > 0 by justifying differentiation under the integral sign. Evaluate Z ∞
0
(e−ax− e−bx)dx x . (5) Prove that
Z ∞ 0
sin(tx)
x(x2+ 1)dx = π
2(1 − e−t) for t > 0 by justifying differentiation under the integral sign.
1
