55.6. EXERCISES 1765

10. Suppose Ω is a simply connected region and u is a real valued function defined on Ω

such that u is harmonic. Show there exists an analytic function, f such that u = Re f .Show this is not true if Ω is not a simply connected region. Hint: You might use theRiemann mapping theorem and Problems 8 and 9. For the second part it might begood to try something like u(x,y) = ln

(x2 + y2

)on the annulus 1 < |z|< 2.

11. Show that w = 1+z1−z maps {z ∈ C : Imz > 0 and |z|< 1} to the first quadrant,

{z = x+ iy : x,y > 0} .

12. Let f (z) = az+bcz+d and let g(z) = a1z+b1

c1z+d1. Show that f ◦g(z) equals the quotient of two

expressions, the numerator being the top entry in the vector(a bc d

)(a1 b1c1 d1

)(z1

)and the denominator being the bottom entry. Show that if you define

φ

((a bc d

))≡ az+b

cz+d,

then φ (AB) = φ (A)◦φ (B) . Find an easy way to find the inverse of f (z) = az+bcz+d and

give a condition on the a,b,c,d which insures this function has an inverse.

13. The modular group2 is the set of fractional linear transformations, az+bcz+d such that

a,b,c,d are integers and ad− bc = 1. Using Problem 12 or brute force show thismodular group is really a group with the group operation being composition. Alsoshow the inverse of az+b

cz+d is dz−b−cz+a .

14. Let Ω be a region and suppose f is analytic on Ω and that the functions fn are alsoanalytic on Ω and converge to f uniformly on compact subsets of Ω. Suppose f isone to one. Can it be concluded that for an arbitrary compact set, K ⊆ Ω that fn isone to one for all n large enough?

15. The Vitali theorem says that if Ω is a region and { fn} is a uniformly bounded se-quence of functions which converges pointwise on a set, S ⊆ Ω which has a limitpoint in Ω, then in fact, { fn} must converge uniformly on compact subsets of Ω toan analytic function. Prove this theorem. Hint: If the sequence fails to converge,show you can get two different subsequences converging uniformly on compact setsto different functions. Then argue these two functions coincide on S.

16. Does there exist a function analytic on B(0,1) which maps B(0,1) onto B′ (0,1) ,the open unit ball in which 0 has been deleted?

2This is the terminology used in Rudin’s book Real and Complex Analysis.