Download Complex Analysis 4 - Exercises - Mathematics and more Exercises Mathematics in PDF only on Docsity!
Homework Assigned on April 9, 2009 due April 14, 2009 (numbering of problems continued from the last assignment with the same due date)
Problem 4 (from Stein & Shakarchi, p.251, #13). The pseudo-hyperbolic distance between two points z, w ∈ D is defined by
ρ(z, w) =
z − w 1 − wz¯
(a) Prove that if f : D → D is holomorphic, then
ρ (f (z), f (w)) ≤ ρ(z, w) for all z, w ∈ C.
Moreover, prove that if f is an automorphism of D then f preserves the pseudo-hyperbolic distance
ρ (f (z), f (w)) = ρ(z, w) for all z, w ∈ C.
Hint: Consider the automorphism
ψα(z) =
z − α 1 − αz¯
and apply the Scharz lemma to ψf (w) ◦ f ◦ ψ− w 1.
(b) Prove that |f ′(z)| 1 − |f (z)|^2
1 − |z|^2
for all z ∈ D.
This is called the Schwarz-Pick lemma. Hint: Consider the limiting case of w → z in Part(a).
Problem 3 (from Stein & Shakarchi, p.252, #17). If
ψα(z) =
α − z 1 − αz¯
for |α| < 1 ,
prove that
1 π
D
|ψ′ α|^2 dxdy = 1 and
π
D
|ψ α′| dxdy =
1 − |α|^2 |α|^2
log
1 − |α|^2
where in the case α = 0 the expression on the right is understood as the limit as |α| → 0.
Hint: The first integral can be evaluated without a calculation. For the second, use polar coordinates, and for each fixed r use contour integration to evaluate the integral in θ.
Problem 5 (on replacing the square-root map in the proof of the Riemann mapping theorem by the
m
-th power map for any integer m ≥ 2 ). Let m ≥ 2 be an integer. Let Ω be a simply connected domain in C. Let z 0 ∈ Ω. Let w = f (z) be a holomorphic univalent map from Ω to the open unit disk ∆ (not necessarily surjective) with f (z 0 ) = 0. Assume that a ∈ ∆ is not in the image of f. Let g(z) be any branch of
( f (z) − a 1 − ¯af (z)
) (^) m^1
defined on Ω, which exists because Ω is simply connected and a ∈ ∆ is not in the image of f. Let
h(z) =
g(z) − g (z 0 ) 1 − g (z 0 )g(z)
Show that h(z) defines a holomorphic univalent map from Ω to ∆ (not nec- essarily surjective) with h (z 0 ) = 0 and that |h′^ (z 0 )| > |f ′^ (z 0 )|.
Problem 6. Let f (z) = z^4 + z^3 + 4z^2 + 2z + 3. Use the argument principle to verify hat f (z) has no zeroes in the first quadrant by showing that
(i) the argument arg f (z) of f (z) is unchanged as z goes along the real axis from the origin to any positive number, and
(ii) when z in the first quadrant goes around in the counterclockwise sense a quarter of the circle of radius R > 0 centered at 0, the argument arg f (z) of f (z) increases by 2π when R is sufficiently large with an error of the order (^) R^1 , and
(iii) when z goes along the positive imaginary axis from Ri to the origin,
arg f (z) = tan−^1
−y^3 + 2y y^4 − 4 y^2 + 3
decrease by 2π when R is sufficiently large with an error of the order 1 R.
