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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
“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
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
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
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.
If z0 é D as well, we say that f is continuous at z0 if lim f(z) = f(z0).
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
f at z0 to be
f(z) - f(z0)
f'(z0) = lim
zÆz0 z - z0
f is called analy...
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