We say that a function f (z) dened in a region U is analytic in a point
z0 if there exists r such that f (z) can be expressed as a convergent power
series an (z z0 )n in the disk Dr (z0 ) U (where Dr (z0 ) is a disk
of radius r around a p
Conformal Metric, Geodesics
Denition 1.1. Let U C be a region, : U R, (z) > 0 for all z U .
Then (z)|dz| is called a conformal metric.
Denition 1.2. Let z1 , z2 U . The curve : [0, 1] U , g(0) = z1 ,
g(1) = z2 is called a geodesic of
1. Let f : U C be a holomorphic function. Let Rf be its Riemann
surface. Let : Rf C be the natural projection. Construct an
example of a function f (or show that it exists) such that
(a) Im = D and is a bijection to its image;
(b) Im = C and
A holomorphic bijective self-map f : U U is called a conformal
Let U C. We say that (z)|dz|, where (z) > 0, is a conformal
Let z1 , z2 U . The curve : [0, 1] U , g(0) = z1 , g(1) = z2 is called a
geodesic of the metric
Theorem: A family of locally uniformly bounded functions is normal.
1. Let f : D D be a holomorphic map. Let dD be the hyperbolic
distance on D. Prove that
dD (f (z1 ), f (z2 ) dD (z1 , z2 ).
Hint: Use Schwarz lemma.
2. (a) We say that M =
Let f : C C be a rational function. Let (z, w) be chordal distance
on the Riemann sphere. We say that a point z belongs to Fatou set
F (f ) if for there exists such that (w, z) < implies
(f n (w), f n (z) <
for all n.
The compliment of the Fa
We dene the singularity of f (z) at to be the singularity at 0 of
g(z) = f ( z ).
1. How many roots does the equation 3z 5 + 21z 4 + 5z 3 + 6z + 7 = 0 have
in D? How many in cfw_z : 1 < |z| < 2? (Hint: use Rouchs Theorem)
2. (a) Let f be
Let > 0. Assume that f (z) is an analytic function in Uz0 := cfw_z :
0 < |z z0 | < . We say that z0 is a removable singularity if there
exists g(z), analytic in cfw_z : |z z0 | < such that g(z) = f (z) for all
z Uz0 .
You can use the follow
Unit disk D = cfw_|z| < 1.
Upper half plane H = cfw_Im z > 0.
1. (a) Let a D, [0, 2). Let Ta, = ei 1z . Check that each Ta,
is an analytic bijective map from D to D.
(b) Let T : D D be an analytic bijective map, such that T (0) = 0,
We say that U is a region if it is an open connected set.
We say that a function f (z) is analytic in a region U if for every point
z0 U there exists r such that f (z) can be expressed as a convergent
power series an (z z0 )n in the disk Dr
1. Find all complex solutions of the equation z 5 = 1. Draw them on the
2. Consider a map f : C\cfw_0 C\cfw_0, f (z) = z . Find the image of (a)
the interior of the unit circle, (b) the exterior of the unit circle. Prove
List of topics for the nal.
1. Geometry of complex numbers. (Section I.1, Marshals notes)
2. Analytic functions. Local properties. (Chapter II, Marshals notes).
3. Maximum modulus principle, Schwartz lemma, Liouvilles Theorem.
(Chapter III, Marshals notes