720 CHAPTER 35. ANALYTIC FUNCTIONS

1. Suppose you have U ⊆ C an open set and f : U → C is analytic but has only realvalues. Find all possible f with these properties.

2. Suppose f is an entire function (analytic on C) and suppose Re f is never 0. Showthat f must be constant. Hint: Consider U = {(x,y) : Re f (x,y)> 0} ,V = {(x,y) : Re f (x,y)< 0} .These are open and disjoint so one must be empty. If V is empty, consider 1/e f (z).Use Liouville’s theorem.

3. Suppose f : C→ C is analytic. Suppose also there is an estimate

| f (z)| ≤M(1+ |z|α

),α > 0

Show that f must be a polynomial. Hint: Consider the formula for the derivative inwhich γr is positively oriented and a circle or radius r for r very large centered at 0,

f (n) (z) =n!

2πi

∫γr

f (w)

(w− z)n+1

and pick large n. Then let r→ ∞.

4. Define for z ∈C sinz≡∑∞k=0 (−1)n z2n+1

(2n+1)! . That is, you just replace x with z. Give asimilar definition for cosz, and ez. Show that the series converges for sinz and that acorresponding series converges for cosz. Then show that

sinz =eiz− e−iz

2i, cosz =

eiz + e−iz

2

Show that it is not longer true that the functions sinz,cosz must be bounded in abso-lute value by 1. Hint: This is a very easy problem if you use the theorem about thezeros of an analytic function, Theorem 35.6.9.

5. Verify the identities cos(z−w) = coszcosw+sinzsinw and similar identities. Hint:This is a very easy problem if you use the theorem about the zeros of an analyticfunction, Theorem 35.6.9.

6. Consider the following contour in which the large semicircle has radius R and thesmall one has radius r ≡ 1/R.

x

y

The function z→ eiz

z is analytic on the curve and on its inside. Therefore, the con-tour integral with respect to the given orientation is 0. Use this contour and theCauchy integral theorem to verify that

∫∞

0sinz

z dz = π/2 where this improper integralis defined as

limR→∞

∫ R

−1/R

sinzz

dz

The function is actually not absolutely integrable and so the precise description ofits meaning just given is important. To do this, show that the integral over the large