56.3. EXERCISES 1783

4. Suppose Ω is a region (connected open set) and f is an analytic function defined onΩ such that f (z) ̸= 0 for any z ∈ Ω. Suppose also that for every positive integer, nthere exists an analytic function, gn defined on Ω such that gn

n (z) = f (z) . Show thatthen it is possible to define an analytic function, L on f (Ω) such that eL( f (z)) = f (z)for all z ∈Ω.

5. You know that φ (z) ≡ z−iz+i maps the upper half plane onto the unit ball. Its inverse,

ψ (z) = i 1+z1−z maps the unit ball onto the upper half plane. Also for z in the upper

half plane, you can define a square root as follows. If z = |z|eiθ where θ ∈ (0,π) , letz1/2 ≡ |z|1/2 eiθ/2 so the square root maps the upper half plane to the first quadrant.Now consider

z→ exp

(−i log

[i(

1+ z1− z

)]1/2). (56.3.13)

Show this is an analytic function which maps the unit ball onto an annulus. Is itpossible to find a one to one analytic map which does this?