We continued the example of \int_0^\infty \sin x/x\,dx from the previous lecture.
Lemma (Jordan). Suppose f is analytic on the closed upper half plane excluding some disc, \left\{z : \operatorname*{Im}z\ge 0\right\}\cap \left\{z : |z| \ge R_0\right\}, which satisfies |f(z)| \le M/R^\beta for some M, \beta > 0 on the outside arc of \Gamma_R = \left\{Re^{i\theta} : R > R_0, 0 \le \theta \le \pi\right\}. Then, for all \alpha > 0, \lim_{R \to \infty}\int_{\Gamma_R}e^{i\alpha z}f(z)\,dz = 0. Proof. Fairly straightforward apart from one estimate. \square
§93–94 (8 Ed §86–87) – Rouche’s Theorem
The argument principle says if f is analytic in and on C, a simple closed curve, except possibly for poles inside C, then the rate of change of the argument along the curve is \Delta_C \arg f=2\pi(z\cdot p), where z is the number of zeros inside C counting multiplicity, and p is the number of poles counting sum of orders.
Theorem (Rouché’s theorem). Let f and g be analytic in and on a simple closed curve C (orientation irrelevant). Suppose |g(z)| < |f(z)| for all z on this curve. Then, f and f+g have the same number of zeros (counting multiplicity) inside C.
For example, (z-i)^2(z+i)^3 has 5 zeros counting multiplicity. Make sure to check conditions before applying theorems.
Example: How many zeros of h(z) = z^7-4z^3-1 lie inside the unit circle? Let f(z) = -4z^3 and g(z) = z^7+z-1, noting that both are polynomials and entire. In general for picking f and g with an annulus, take the larger power outside the annulus. This case is a bit more special. First, we have |f|=4 on C and by the triangle inequality, |g(z)| \le |z^7| + |z| + |-1| =3< 4=|f(z)|. Because |g| < |f| on C, Rouché’s theorem implies that f and f+g=h have the same number of zeros inside C, namely 3 because f has a 3-fold zero at 0.