Chapter 2 Complex analysis 2.1 Basics Given z0 2 C, r > 0 we define D(z0; r) := fz 2 C : jz − z0j < rg (open disk), D(z0; r) := fz 2 C : jz − z0j 6 rg (closed disk). A subset U ⊆ C is called open if U = ; or if for every z0 2 C there is δ > 0 with c D(z0; δ) ⊂ U. A subset U ⊆ C is called closed if U := C n U is open. It is known that a subset of C is compact (i.e., every open cover of the set has a finite subcover) if and only if it is closed and bounded (see the Prerequisites). In what follows, U is a non-empty open subset of C and f : U ! C a function. We say that f is holomorphic or analytic in z0 2 U if f(z) − f(z ) lim 0 exists. z!z0 z − z0 0 In that case, the limit is denoted by f (z0). We say that f is analytic on U if f is analytic in every z 2 U; in that case, the derivative f 0(z) is defined for every z 2 U. We say that f is analytic around z0 if it is analytic on some open disk D(z0; δ) for some δ > 0. Finally, given a not necessarily open subset A of C and a function f : A ! C, we say that f is analytic on A if there is an open set U ⊇ A such that f is defined on U and analytic on U. An everywhere analytic function f : C ! C is called entire. 35 For any two analytic functions f; g on some open set U ⊆ C,we have the usual rules for differentiation (f ±g)0 = f 0±g0,(fg)0 = f 0g+fg0 and (f=g)0 = (gf 0−fg0)=g2 (the latter is defined for any z with g(z) 6= 0). Further, given a non-empty set U ⊆ C, and analytic functions f : U ! C, g : f(U) ! C, the composition g ◦ f is analytic on U and (g ◦ f)0 = (g0 ◦ f) · f 0. Recall that a power series around z0 2 C is an infinite sum 1 X n f(z) = an(z − z0) n=0 with an 2 C for all n 2 Z>0. The radius of convergence of this series is given by −1 pn R = Rf = lim sup janj : n!1 We state without proof the following fact. P1 n Theorem 2.1. Let z0 2 C and f(z) = n=0 an(z − z0) a power series around z0 2 C with radius of convergence R > 0. Then f defines a function on D(z0;R) (k) which is analytic infinitely often. For k > 0 the k-th derivative f of f has a power series expansion with radius of convergence R given by 1 (k) X n−k f (z) = n(n − 1) ··· (n − k + 1)an(z − z0) : n=k In each of the examples below, R denotes the radius of convergence of the given power series. 1 X zn ez = ;R = 1; (ez)0 = ez: n! n=0 1 X z2n cos z = (eiz + e−iz)=2 = (−1)n ;R = 1; cos 0z = − sin z: (2n)! n=0 1 X z2n+1 sin z = (eiz − e−iz)=2i = (−1)n ;R = 1; sin 0z = cos z: (2n + 1)! n=0 1 X α (1 + z)α = zn;R = 1; ((1 + z)α)0 = α(1 + z)α−1 n n=0 α α(α − 1) ··· (α − n + 1) where α 2 ; = : C n n! 1 X (−1)n−1 log(1 + z) = · zn;R = 1; log 0(1 + z) = (1 + z)−1: n n=1 36 2.2 Cauchy's Theorem and some applications For the necessary definitions concerning paths, line integrals and topology we re- fer to the Prerequisites. In the remainder of this course, a path will always be a piecewise continuously differentiable path. Recall that for a piecewise continuously differentiable path γ, say γ = γ1 + ··· + γr where γ1; : : : ; γr are paths with continu- ously differentiable parametrizations gi :[ai; bi] ! C, and for a continuous function R Pr R bi 0 f : γ ! we have f(z)dz = f(gi(t))g (t)dt. C γ i=1 ai i Theorem 2.2 (Cauchy). Let U ⊆ C be a non-empty open set and f : U ! C an analytic function. Further, let γ1; γ2 be two paths in U with the same start point and end point that are homotopic in U. Then Z Z f(z)dz = f(z)dz: γ1 γ2 Proof. Any textbook on complex analysis. Corollary 2.3. Let U ⊆ C be a non-empty, open, simply connected set, and f : U ! C an analytic function. Then for any closed path γ in U, I f(z)dz = 0: γ Proof. The path γ is homotopic in U to a point, and a line integral along a point is 0. Corollary 2.4. Let γ1; γ2 be two contours (closed paths without self-intersections traversed counterclockwise), such that γ2 is contained in the interior of γ1. Let U ⊂ C be an open set which contains γ1; γ2 and the region between γ1 and γ2. Further, let f : U ! C be an analytic function. Then I I f(z)dz = f(z)dz: γ1 γ2 Proof. 37 Let z0; z1 be points on γ1; γ2 respectively, and let α be a path from z0 to z1 lying in- side the region between γ1 and γ2 without self-intersections. Then γ1 is homotopic in U to the path α + γ2 − α, which consists of first traversing α, then γ2, and then α in the opposite direction. Hence I Z I Z I f(z)dz = + − f(z)dz = f(z)dz: γ1 α γ2 α γ2 Corollary 2.5 (Cauchy's Integral Formula). Let γ be a contour in C, U ⊂ C an open set containing γ and its interior, z0 a point in the interior of γ, and f : U ! C an analytic function. Then 1 I f(z) · dz = f(z0): 2πi γ z − z0 Proof. Let γz0,δ be the circle with center z0 and radius δ, traversed counterclockwise. Then by Corollary 2.4 we have for any suf- ficiently small δ > 0, 1 I f(z) 1 I f(z) · dz = · dz: 2πi z − z0 2πi z − z0 γ γz0,δ 38 Now, since f(z) is continuous, hence uniformly continuous on any sufficiently small compact set containing z0, I I 1 f(z) 1 f(z) · dz − f(z0) = · dz − f(z0) 2πi z − z0 2πi z − z0 γ γz0,δ Z 1 2πit f(z0 + δe ) 2πit = 2πit · δe dt − f(z0) 0 δe Z 1 2πit 2πit = f(z0 + δe ) − f(z0) dt 6 sup jf(z0 + δe ) − f(z0)j 0 06t61 ! 0 as δ # 0: This completes our proof. We now show that every analytic function f on a simply connected set has an anti-derivative. We first prove a simple lemma. Lemma 2.6. Let U ⊆ C be a non-empty, open, connected set, and let f : U ! C be an analytic function such that f 0 = 0 on U. Then f is constant on U. Proof. Fix a point z0 2 U and let z 2 U be arbitrary. Take a path γz in U from z0 to z which exists since U is (pathwise) connected. Then Z 0 f(z) − f(z0) = f (w)dw = 0: γz Corollary 2.7. Let U ⊂ C be a non-empty, open, simply connected set, and f : U ! C an analytic function. Then there exists an analytic function F : U ! C with F 0 = f. Further, F is determined uniquely up to addition with a constant. 0 0 Proof (sketch). If F1;F2 are any two analytic functions on U with F1 = F2 = f, then 0 0 F1 − F2 is constant on U since U is connected. This shows that an anti-derivative of f is determined uniquely up to addition with a constant. It thus suffices to prove the existence of an analytic function F on U with F 0 = f. 39 Fix z0 2 U. Given z 2 U, we define F (z) by Z F (z) := f(w)dw; γz where γz is any path in U from z0 to z. This does not depend on the choice of γz. For let γ1; γ2 be any two paths in U from z0 to z. Then γ1 − γ2 (the path consisting of first traversing γ1 and then γ2 in the opposite direction) is homotopic to z0 since U is simply connected, hence Z Z I f(z)dz − f(z)dz = f(z)dz = 0: γ1 γ2 γ1−γ2 F (z+h)−F (z) To prove that limh!0 h = f(z), take a path γz from z0 to z and then the line segment [z; z + h] from z to z + h. Then since f is uniformly continuous on any sufficiently small compact set around z, Z Z Z F (z + h) − F (z) = − f(w)dw = f(w)dw γz+[z;z+h] γz [z;z+h] Z 1 Z 1 = f(z + th)hdt = h f(z) + (f(z + th) − f(z))dt : 0 0 So Z 1 F (z + h) − F (z) − f(z) = (f(z + th) − f(z))dt h 0 6 sup jf(z + th) − f(z)j ! 0 as h ! 0: 06t61 This completes our proof.