MATH 202a Complex Analysis UCSB

7. Examples of Functions defined by Series Here are some interesting examples. Consider the dierential equation y (z) = y. subject to the initial value y(0) = 1, where y is some holomorphic function of z. We posit a solution that is given by a power

HOMEWORK, DUE WEDNESDAY OCTOBER 9TH 1. Show that the function f (x) = 1 exp( x2 ) for x = 0 0 for x = 0 is innitely dierentiable and that f (k) (0) = 0 for every k. Thus f is not analytic. 2. Show that the function g(x) = is innitely dierentiable. 3

26. Infinite Products Denition 26.1. Let p1 , p2 , . . . be an innite sequence of complex numbers. p1 p2 p3 . . . pn = i=1 pi is dened to be the limit of the partial products Pn = p 1 , p 2 , . . . , p n . Note that we have to be very careful i

13. Complex Integration First some basic denitions. Denition 13.1. Let X be a topological space. A curve in X is a continuous function : [a, b] X. There are obvious notions attached to this denition; the image (aka trace), endpoints, writing a curv

1. (15pts) Write down the Cauchy-Riemann equations (the complex or real form as you wish) and show that any holomorphic function must satisfy them. Under what conditions is it true that a function which satises the Cauchy-Riemann equations is holomor

MODEL ANSWERS TO THE SIXTH HOMEWORK 1. Let f (z) = 6z 3 and g(z) = z 7 2z 5 + 6z 3 z + 1. Then, |f (z)| = 6 on the circle |z| = 1 and |g(z) f (z)| |z 7 | + |2z 5 | + |z| + |1| = 5 on |z| = 1. Thus by Rouch, f and g have the same number of zeroes

10. Circles and lines Back to the cross-ratio. Suppose we have z1 , z2 , z3 , z4 . I claim that the cross-ratio is real i these four points lie on a circle. Indeed the cross-ratio is equal to z2 z 4 z1 z 3 . z2 z 3 z1 z 4 Taking arguments, we get

1. Compare and contrast The theory of functions of one real variable holds many nasty surprises, which in contrast to the theory of functions of one complex variable, might even be considered pathological. For example it is easy to write down a funct

19. Homotopy Version of Cauchys Results Denition 19.1. Let f : X Y and g : X Y be two continuous map of topological spaces. A homotopy between f and g is a continuous map F : X I Y, such that F |X{0} = f and F |X{1} = g. If further we are given a