文档介绍：Section
 ¢¡£¡¥¤§¦¨¤§©
¤
 £©
Complex-Valued Functions:
 ¢¡£¡¥¤§¦¨¤§©  £
 £© Motion in the Plane
In Section we considered the problem of extending the elementary functions of calculus
plex-valued functions of plex variable, while at the same time extending many
of the concepts of the first six chapters to these new functions. In this section we will
plex-valued functions of a real variable, that is, functions of the form f : R →
C. Such functions are often used to describe the motion of an object in the plane; if we
think of the real variable t as measuring time, then we may interpret f(t) as the location
of an object in plex plane at time t.
Since limits are at the foundation of most concepts in calculus, we begin with a defi-
nition of limit in this setting.
Definition Suppose f : R → C and f is defined for all t in an interval about the point
a. We say that the limit of f(t) as t approaches a is L, denoted
lim f(t) = L,
t→a
if whenever {tn} is a sequence of real numbers with tn =6 a for all n and
lim tn = a,
n→∞
then
lim f(tn) = L.
n→∞
Suppose f : R → C. If we let x(t) = <(f(t)) and y(t) = =(f(t)), then
f(t) = x(t) + iy(t).
Hence, from our work in Section ,
lim f(t) = lim x(t) + i lim y(t). ()
t→a t→a t→a
This result also holds if we modify our definition of limit to include one-sided limits and
limits to ∞ or −∞.
Example Suppose a particle moves in the plane so that its position at time t is given
by
f(t) = cos(2πt) + i sin(2πt) = e2πit.
1
plex-Valued Functions: Motion in the Plane Section
t =
t = t = 0, 1
t =
Figure Motion of a particle on the unit circle centered at the origin
If we let C denote the unit circle centered at the origin, then f(t) is a point in plex
plane on C, 2πt units from (1, 0) in the counterclockwise direction along the circumference
1
of C. For example, at time t = 0 the particle is at f(0) = 1,