Complex Analysis Minh-Tam Trinh Note. I Am Giving the Second Lecture
WOMP 2019: Complex Analysis
Minh-Tam Trinh
Note. I am giving the second lecture; Meg Doucette is giving the first.
1. Functions Holomorphic functions have a very idiosyncratic geometry. I will explain how some major theorems illustrate this. Henceforth, let C be an open domain, D an open disk, and @D the   positively-oriented loop formed by the boundary of D. Let f C be holomorphic. The residue formula shows that if f =f has no zeros/poles on W ! 0 @D, then Z 1 f 0 X f X (1.1) dz resz. 0 / ordz.f /: 2i f D f D @D f 0 zeros z of f poles z of f
(Above, we used the fact that any zero of f of order n corresponds to a simple pole of f 0=f of residue n.) At the same time, the substitution w f .z/ shows that D 1 Z f .z/ 1 Z dw (1.2) 0 dz winding number of f .@D/ around 0: 2i @D f .z/ D 2i f .@D/ w D In short, the number of zeros of f in D, counting multiplicities, is given by a winding number! This result is known as Cauchy’s argument principle. The following result says that if you perturb a function by a very small amount in a fixed neighborhood, then its zeros can move around, but their total multiplicity is conserved. Theorem 1.1 (Rouché). If h C is holomorphic and h < f on @D, then f h has the W ! j j j j C same number of zeros as f in D.
Proof. Let ft .z/ f .z/ th.z/. The winding number of ft can’t change discontinuously as t runs D C through the interval Œ0; 1. Let’s use Rouché to prove the fundamental theorem of algebra. We claim that if
n n 1 (1.3) f z an 1z a1z a0; D C C C C P i then f has n zeros in C. Indeed, in the statement of Rouché, let h.z/ ai z and pick a D 0 i n 1 disk D so big that zn > h.z/ on the boundary @D. Ä Ä j j j j Corollary 1.2 (Open Mapping). Any nonconstant holomorphic function is an open map.
Proof. Using Rouché, show that if f .z/ 0, then D.0/ f .Dı .z// for some ı > 0 and D Â > 0. Corollary 1.3 (Maximum Modulus). If f C is nonconstant, then f cannot attain a W ! j j maximum on . Proof. By the open mapping theorem, any z can be perturbed to a nearby value w such 2 2 that f .w/ > f .z/ . j j j j Complex Analysis 2
2. Spaces Let’s use the open mapping theorem to give a second proof of the fundamental theorem of algebra. This proof will involve a new space, namely, the Riemann sphere
(2.1) C C : O D [ f1g Let f .z/ be a polynomial. As a map C C, it extends to a map C C that fixes . Since C is ! O ! O 1 O compact and Hausdorff, the image of fO is a closed set. But by the open mapping theorem, it is also open. Since C is connected, we deduce that f .C/ C, whence f .C/ C. O O D O D Theorem 2.1. Every connected, simply-connected Riemann surface is conformally equivalent to one of the following: (1) The plane C. (2) The Riemann sphere C. O (3) The open unit disk D. These options are conformally distinct.
We can learn a lot of mathematics simply by computing the conformal automorphism groups of C; C; D. Let’s do them in that order. O Theorem 2.2. Aut.C/ is the group of affine transformations z az b. 7! C Proof. Conformal automorphisms are isometries, so for C, decompose into the composition of a translation and a homothety (i.e., rotation + dilation).