Math144Notes

# Limits and derivatives of complex valued functions

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e “winding” maps. (c) The function f: C ÆC given by f(z) = 1/z = z-1 is a special case of (a) above, and II “winds” the unit circle backwards. It maps the circle of radius r backwards around the circle of radius -r. (d) The function f: C ÆC given by f(z) = z– agrees with 1/z on the unit circle, but not II elsewhere. Limits and Derivatives of Complex-valued Functions Definition 2.4 If D ¯ C then a point z0 not necessarily in D is called a limit point of D if I every neighborhood of z0 contains points in D other than itself. Illustrations in class Definition 2.5 Let f: DÆC and let z0 be a limit point of D. Then we say that f(z) Æ L as I z Æ z0 if for each œ &gt; 0 there is a © &gt; 0 such that |f(z) - L) &lt; œ whenever 0 &lt; |z - z0| &lt; œ. When this happens, we also write lim f(z) = L. zÆz0 If z0 é D as well, we say that f is continuous at z0 if lim f(z) = f(z0). zÆz0 Fact: Every closed-form (single-valued) function of a complex variable is continuous on its domain. 6 Definition 2.6 Let f: DÆC and let z0 be in the interior of D. We define the derivative of I f at z0 to be f(z) - f(z0) f'(z0) = lim zÆz0 z - z0 f is called analy...
View Full Document

## This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

Ask a homework question - tutors are online