Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on January 29, 2009
due February 5, 2009
Problem 1 (from Stein & Shakarchi, p.26, #7). The family of mappings in-
troduced here plays an important rˆole in complex analysis. These mappingsk
sometimes called Blaschke factors, will reappear in various applications later.
(a) Let z, w be two complex numbers such that ¯zw 6= 1. Prove that
¯
¯
¯
¯
w−z
1−¯wz
¯
¯
¯
¯
<1 if |z|<1 and |w|<1,
and also that ¯
¯
¯
¯
w−z
1−¯wz
¯
¯
¯
¯
= 1 if |z|= 1 or |w|= 1.
Hint: Why can one assume that zis real? It then suffices to prove that
(r−w)(r−¯w)≤(1 −rw)(1 −r¯w)
with equality for appropriate rand |w|.
(b) Prove that for a fixed win the unit disk D, the mapping
F:z7→ w−z
1−¯wz
satisfies the following conditions:
(i) Fmaps the unit disc to itself (that is, F:D→D), and is complex-
differentiable at every point of the unit disk D.
(ii) Finterchanges 0 and w, namely F(0) = wand F(w) = 0.
(iii) |F(z)|= 1 if |z|= 1.
(iv) F:D→Dis bijective. [Hint: Calculate F◦F.]
Problem 2 (from Stein & Shakarchi, p.27, #9). Consider the polar coordi-
nates (r, θ) so that x=rcos θand y=rsin θwhich can also be written as
z=x+iy =r(cos θ+isin θ). Show that in polar coordinates (r, θ), the
Cauchy-Riemann equations for the function f=u+iv take the form
∂u
∂r =1
r
∂v
∂θ and 1
r
∂u
∂θ =−∂v
∂r .
