276 CHAPTER 11. FUNCTIONS OF ONE COMPLEX VARIABLE

32. Use Problem 30 to state a theorem whose conclusion is that

f (α)n(γ∗,α) =1

2πi

∫γ∗

f (z)z−α

dz.

Here γ∗ is a closed curve. This is a case of a general Cauchy integral formula forcycles.

33. Using the open mapping theorem, show that if U is an open connected set and f isanalytic and | f | has a maximum on U , then this maximum occurs on the boundaryof U . This is called the maximum modulus theorem.

34. Use the counting zeros theorem, get an estimate for a ball centered at 0 which willcontain all zeros of p(z) = z7 + 5z5− 3z2 + z+ 5. Hint: You might compare withq(z) = z7. You know all of its zeros. For |z| = r large enough, consider λ z7 +(1−λ ) p(z) and λ ∈ [0,1].

35. Use the counting zeros theorem to give another proof of the fundamental theorem ofalgebra. This one is even easier than the earlier one based on Liouville’s theorem,but it uses the harder theorem about counting zeros.

36. As in the Liouville proof of the fundamental theorem of algebra, if p(z) is a non-constant polynomial with no zero, then lim|z|→∞ (1/ |p(z)|) = 0. Then 1/ |p(z)| hasa maximum on C, say it has such a maximum at z0. Now exploit the maximummodulus theorem of Problem 33 on balls containing z0 at the center to obtain a con-tradiction. Provide details.

37. Rouche’s theorem considers the case where there are no poles or zeros on C (z0,r) ,the counter clockwise oriented circle bounding B(z0,r) and finitely many zeros andpoles in B(z0,r). Such a function is called meromorphic. Rouche’s theorem says thatin the situation just described and f such a meromorphic function 1

2πi∫

C(z0,r)f ′(z)f (z) dz=

N−P where N is the number of zeros counted according to multiplicity and P thenumber of poles, also counted according to something which will be apparent whenthis is proved. Use the proof given above for counting zeros to verify this moregeneral theorem. It just involves finding the residues of f ′/ f at the zeros and poles.

38. If U is an open connected subset of C and f : U → R is analytic, what can you sayabout f ? Hint: You might consider the open mapping theorem.

39. Using the fundamental theorem of algebra and the partial fractions theorem of Prob-lem 19 on Page 39, show that the result of Theorem 11.3.2 is always obtained forrational functions. Give an example where Theorem 11.3.2 is better.

40. Show that if U is an open connected set in C and f : U → C is one to one andanalytic, then f−1 : f (U)→ U is also analytic. Really? What about f (z) = z3?Hint: Do something like in Theorem 7.10.1. Here it may be easier because by theopen mapping theorem, you know f (U) is open so there are no endpoints to worryabout.