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IV.3. Zeros of an Analytic Function
Note. We now explore factoring series in a way analogous to factoring a poly- nomial. Recall that if p is a polynomial with a zero a of multiplicity m, then p(z) = (z − a)mt(z) for a polynomial t(z) such that t(a) 6 = 0.
Definition. If f : G → C is analytic and a ∈ G satisfies f (a) = 0, then a is a zero of multiplicity m ≥ 1 if there is analytic g : G → C such that f (z) = (z − a)mg(z) where g(a) 6 = 0.
Note. “The reader might be pleasantly surprised to know that after many years of studying Mathematics [they are] right now on the threshold of proving the Fun- damental Theorem of Algebra.” (page 76)
Definition. An entire function is a function analytic in the entire complex plane. Entire functions are sometimes called integral functions.
Note. An area of study in complex analysis is entire function theory. A classical book in this area is Ralph Boas’ Entire Functions (Academic Press, 1954). Results in this are often concern factorization and rates of growth (see Conway’s Chapter XI).
Note/Proposition IV.3.3. If f is an entire function then f (z) =
∑^ ∞
n=
anzn^ with
infinite radius of convergence.
Theorem IV.3.4. Liouville’s Theorem. If f is a bounded entire function then f is constant.
Proof. Suppose |f (z)| ≤ M for all z ∈ C. By Cauchy’s Estimate (Corollary IV.2.14) with n = 1, |f ′(z)| ≤ M/R for any disk B(z; R). Since f is entire, the inequality holds for all R and with R → ∞ we see that f ′(z) = 0 for all z ∈ C. Therefore, f is a constant function by Proposition III.2.10.
Note. Notice the quote on page 77! “The reader should not be deceived into thinking that this theorem is insignificant because it has such a short proof. We have expended a great deal of effort building up machinery and increasing our knowl- edge of analytic functions. We have plowed, planted, and fertilized; we shouldn’t be surprised if, occasionally, something is available for easy picking.” Of course, Liouville’s Theorem does not hold for functions of a real variable: Con- sider sin x, cos x, 1/(x^2 + 1).
Note. As you will see in Field Theory, there is no purely algebraic proof of the Fundamental Theorem of Algebra. One proof (the one from my graduate algebra class) uses the fact that an odd degree real polynomial has a real zero. That is, it uses the Intermediate Value Theorem from analysis. We now give a proof of the Fundamental Theorem of Algebra which is analytic (i.e., uses analysis) and is based primarily on Liouville’s Theorem.
Theorem IV.3.7. Let G be a connected open set and let f : G → C be analytic. The following are equivalent.
(a) f ≡ 0 on G,
(b) there is a point a ∈ G such that f (n)(a) = 0 for all n ∈ Z, n ≥ 0, and
(c) the set {z ∈ G | f (z) = 0} has a limit point in G.
Note. Theorem IV.3.7 does not hold for functions of a real variable (where we take “analytic” to mean continuously differentiable). Recall that
f (x) =
e−^1 /x^2 , x > 0 0 , x ≤ 0
is infinitely differentiable for all x ∈ R and f (n)(0) = 0 for all n ∈ Z, n ≥ 0, but f 6 ≡ 0 on R (so (b) does not imply (a)). Recall g(x) =
x^2 sin(1/x), x 6 = 0 0 , x = 0 has zeros {x ∈ R | x = 1/(nπ), n ∈ Z} ∪ { 0 }. So g is continuously differentiable on R and the set of zeros has a limit point, but g 6 ≡ 0 on R. That is, (c) does not imply (a).
Corollary IV.3.8. If f and g are analytic on a region G (where G is an open connected set), then f ≡ g if and only if {z ∈ G | f (z) = g(z)} has a limit point in G.
Note. Corollary IV.3.8 does not hold in R. Consider f (x) ≡ 0 and
g(x) =
x^2 sin(1/x), x 6 = 0 0 , x = 0.
Note. Theorem IV.3.7 allows us to factor analytic functions as given in the fol- lowing.
Corollary IV.3.9. If f is analytic on an open connected set G and f is not identically zero then for each a ∈ G with f (a) = 0, there is n ∈ N and an analytic function g : G → C such that g(a) 6 = 0 and f (z) = (z − a)ng(z) for all z ∈ G. That is, each zero of f has finite multiplicity.
Corollary IV.3.10. If f : G → C is analytic and not constant, then for any a ∈ G where f (a) = 0, there is an R > 0 such that B(a; R) ⊆ G and f (z) 6 = 0 for 0 < |z − a| < R.
Note. The following is extremely important! At least, I often use it in my complex research.
Recall. Exercise III.3.17 says: “Let G be a region and suppose that f : G → C is analytic such that f (G) is a subset of a circle or a line. Then f is constant.” Now for the the Maximum Modulus Theorem.
