Math 752 Spring 2015 Class 2/11 1 Winding numbers Theorem 1 (Cauchy's integral formula). G ⊆ C open, f : G ! C analytic, γ closed path in G. If n(γ; w) = 0 for all w in the complement of G, then for a 2 Gnfγg 1 Z f(z) n(γ; a)f(a) = dz: 2πi γ z − a Remark 1. The condition n(γ; w) = 0 is crucial since it prohibits γ winding about holes outside the domain of f. R Proof. By definition of winding number we need to show that f(z) γ(w − −1 R −1 z) dw = γ f(w)(w − z) dw for z 'inside' γ. We consider their difference and show that it defines a bounded entire function (which by Liouville's theorem is constant). If n(γ; z) = 0 for z 2 G (make a sketch of examples to understand this R −1 condition), then f(z) γ(w − z) dw = 0. Hence for such z Z f(w) − f(z) Z f(w) dw = dw: (1) γ w − z γ w − z We note that the integral on the left defines an analytic function in all f(w)−f(z) of G (since for every w the function z 7! w−z can be continued to an analytic function by setting its value to f 0(z) at z = w), and the integral on the right defines an analytic function on the complement of γ (differentiation may be performed under the integral sign), in particular on the complement of G. Let H = fw 2 C : n(γ; w) = 0g. Consider (R f(w)−f(z) γ w−z dw if z 2 G; F (z) = R f(w) γ w−z dw if z 2 H: It follows from (1) that F is well-defined (This is an issue since G \ H 6= ;), and it is entire (note that C = H [ G). Since γ is closed, it is compact, and the right hand side goes uniformly to zero as jzj ! 1. Hence f is bounded, and by Liouville's theorem constant. The constant can only be zero by considering limjzj!1 f(z). We get in particular Z 1 Z f(w) f(z) dw = dw γ w − z γ w − z for all z 2 G, which is the claimed identity. This formula extends to sums over closed curves. The winding number P condition can be relaxed to n(γk; w) = 0 for every w in the complement of the union of the paths. The substitution f(z) = g(z)(z − a) in this theorem gives then Theorem 2. Let G be an open subset of the plane and f : G ! C be P analytic. If γ1; :::; γn are closed rectifiable curves in G with k n(γk; w) = 0 for all w the complement of G, then n X Z f(w)dw = 0: k=1 γk Proof. This will be a midterm question. The integral formulas may be differentiated under the integral sign. Theorem 3 (Morera's theorem). G region, f : G ! C continuous so that R for each triangular path Γ in G the identity Γ f = 0 holds. Then f is analytic in G. Proof. May assume that G is a disk with center a. Let [a; z] be the line segment connecting a and z. Define F by Z F (z) = f(w)dw: [a;z] We will show that F 0 = f, and since f is continuous, this will imply that F is analytic. But then also F 0 = f is analytic. The assumed independence of triangular paths implies Z F (z) − F (z0) = f(w)dw [z0;z] and continuity of f implies that F (z) − F (z0) − f(z0) ≤ supfjf(w) − f(z0)j : w 2 [z0; z]g ! 0 z − z0 as z ! z0. 2 Homotopy and simple connectivity We need to investigate under which conditions on the paths γi (and/or G) P the identity n(γi; w) = 0 for w2 = G is satisfied. Definition 1. Let γ0; γ1 : [0; 1] ! G be two closed paths. Then γ0 is homotopic to γ1 if there exists continuous Γ : [0; 1] × [0; 1] ! G so that Γ(:; 0) = γ0; Γ(:; 1) = γ1; and s 7! Γ(s; t) is closed for every t. Notation: γ0 ∼ γ1. Note that ∼ defines an equivalence relation..