Limits and derivatives of complex valued functions

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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 > 0 there is a © > 0 such that |f(z) - L) < œ whenever 0 < |z - z0| < 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...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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