51.8. EXERCISES 1649

51.8 Exercises1. If U is simply connected, f is analytic on U and f has no zeros in U, show there

exists an analytic function, F, defined on U such that eF = f .

2. Let f be defined and analytic near the point a ∈ C. Show that then

f (z) =∞

∑k=0

bk (z−a)k

whenever |z−a|< R where R is the distance between a and the nearest point wheref fails to have a derivative. The number R, is called the radius of convergence andthe power series is said to be expanded about a.

3. Find the radius of convergence of the function 11+z2 expanded about a = 2. Note

there is nothing wrong with the function, 11+x2 when considered as a function of a

real variable, x for any value of x. However, if you insist on using power series, youfind there is a limitation on the values of x for which the power series converges dueto the presence in the complex plane of a point, i, where the function fails to have aderivative.

4. Suppose f is analytic on all of C and satisfies | f (z)| < A+B |z|1/2 . Show f is con-stant.

5. What if you defined an open set, U to be simply connected if C \U is connected.Would it amount to the same thing? Hint: Consider the outside of B(0,1) .

6. Let γ (t) = eit : t ∈ [0,2π] . Find∫

γ1zn dz for n = 1,2, · · · .

7. Show i∫ 2π

0 (2cosθ)2n dθ =∫

γ

(z+ 1

z

)2n ( 1z

)dz where γ (t) = eit : t ∈ [0,2π] . Then

evaluate this integral using the binomial theorem and the previous problem.

8. Suppose that for some constants a,b ̸= 0, a,b∈R, f (z+ ib) = f (z) for all z∈C andf (z+a) = f (z) for all z∈C. If f is analytic, show that f must be constant. Can yougeneralize this? Hint: This uses Liouville’s theorem.

9. Suppose f (z) = u(x,y) + iv(x,y) is analytic for z ∈ U, an open set. Let g(z) =u∗ (x,y)+ iv∗ (x,y) where (

u∗

v∗

)= Q

(uv

)where Q is a unitary matrix. That is QQ∗ = Q∗Q = I. When will g be analytic?

10. Suppose f is analytic on an open set, U, except for γ∗ ⊂U where γ is a one to onecontinuous function having bounded variation, but it is known that f is continuous onγ∗. Show that in fact f is analytic on γ∗ also. Hint: Pick a point on γ∗, say γ (t0) andsuppose for now that t0 ∈ (a,b) . Pick r > 0 such that B=B(γ (t0) ,r)⊆U. Then showthere exists t1 < t0 and t2 > t0 such that γ ([t1, t2])⊆ B and γ (ti) /∈ B. Thus γ ([t1, t2]) isa path across B going through the center of B which divides B into two open sets, B1,