272 CHAPTER 11. FUNCTIONS OF ONE COMPLEX VARIABLE

t ∈ [0,1] . Now consider the function of a real variable f (z0 + t (z− z0)) and considerreal and imaginary parts. You could apply the mean value theorem to these.

6. ↑ Suppose f : C→ C and f ′ exists on C. Such a function is called entire. Supposef is bounded. Then show f must be constant. This is Liouville’s theorem. Notehow different this is than what we see for functions of a real variable. Hint: Pickz. Use formula 11.2 to describe f ′ (z) where γ∗R is a large circle including z on itsinside. Thus, f ′ (z) = 1

2πi∫

γR

f (w)(w−z)2 dw. Now use Theorem 11.1.6 to get an estimate

for | f ′ (z)| which is a constant times 1/R. However, R is arbitrary. Hence f ′ (z) = 0.Now use the above problem.

7. ↑ Show that if p(z) is a non-constant polynomial, then there exists z0 such thatp(z0) = 0. Hint: If not, then 1/p(z) is entire. Just use the quotient rule to see ithas a derivative. Explain why it is bounded and use Liouville’s theorem to assert thatthen it is a constant which it obviously isn’t. This is the shortest known proof of thefundamental theorem of algebra.

8. Let f : C→ C be entire (has a derivative on all of C) and suppose that

max{| f (z)| : |z| ≤ R} ≤CRk.

Then show that f (z) is actually a polynomial of degree k. Hint: Recall the formulafor the derivative in terms of the Cauchy integral.

9. Let D(0,1) be the closed unit disk and let fn be analytic on and near D. Suppose alsothat fn → f uniformly on D(0,1). Show that f is also analytic on B(0,1). If f isan arbitrary continuous function defined on D(0,1), does it follow that there exists asequence of polynomials which converges uniformly to f on D(0,1)? In other words,does the Weierstrass approximation theorem hold in this setting?

10. Suppose you have a sequence of functions { fn} analytic on an open set U . If theyconverge uniformly to a function f , show that f is also analytic on U .

11. For n = 1,2, ... and an complex numbers, the Dirichlet series is ∑∞n=1

anns . Here s is a

complex number. Thus ns ≡ exp(s log(n)) where this refers to the principal branchof the logarithm. Show 1

ns =1

nRes ei ln(n) Im(s). Then show that if the an are uniformlybounded, the Weierstrass M test applies and the Dirichlet series converges absolutelyand uniformly for Res ≥ 1+ ε for any positive ε . Explain why the function of sis analytic for Re(s) > 1. Obtain more delicate results using the Dirichlet partialsummation formula in case ∑n |an|< ∞. Note that if all an = 1, this function is ξ (s)the Riemann zeta function whose zeros are a great mystery.

12. Suppose f ′ exists on C and f (zn) = 0 with zn→ z where the zn are distinct complexnumbers. Show that then f (z) = 0 on C. Hint: If not, then explain why f (w) =(w− z)m g(w) where g(z) ̸= 0. Then 0 = f (zn) = (zn− z)m g(zn) . Hence g(z) =0, a contradiction. Thus argue the power series of f expanded about z is 0 and itconverges to f on all of C.

13. The power series for sinz,cosz converge for all z ∈ C. Using the above problem,explain why the usual trig. identities valid for z,w real continue to hold for all com-plex z,w. For example, sin(z+w) = sin(z)cos(w)+ cos(z)sin(w). If you allow z