文档介绍：Section
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The Calculus plex
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In this section we will discuss limits, continuity, differentiation, and Taylor series in the
context of functions which take plex values. Moreover, we will plex
extensions of a number of familiar functions. plex numbers behave algebraically
like real numbers, most of our results and definitions will look like the analogous results
for real-valued functions. We will avoid going into much detail; plete story of the
calculus plex-valued functions is best left to a course plex analysis. However,
we will see enough of the story to enable us to make effective use plex numbers in
elementary calculations.
We begin with a definition of the limit of a sequence plex numbers.
Definition We say that the limit of a sequence plex numbers {zn} is L, and write
lim zn = L,
n→∞
if for every > 0 there exists an integer N such that
|zn − L| <
whenever n > N.
Notice that the only difference between this definition and the definition of the limit of
a sequence given in Section is the use of the magnitude of plex number in place of
the absolute value of a real number. Even here, the notation is the same. The point is the
same as it was in Chapter 1: the limit of the sequence {zn} is L if we can always ensure
that the values of the sequence are within a desired distance of L by going far enough out
in the sequence.
Now if zn = xn + yni and L = a + bi, then lim zn = L if and only if
n→∞
2 2
lim |zn − L| = lim (xn − a) + (yn − b) = 0,
n→∞ n→∞ p
the latter of which occurs if and only if lim xn = a and lim yn = b. Hence we have the
n→∞ n→∞
following useful result.
Proposition Let zn = xn + yni and L = a + bi. Then
lim zn = L
n→∞
1
2 The Calculus plex Functions Section
if and only if
lim xn = a and lim yn = b.
n→∞ n→∞
Thus to determine the limiting behavior of a sequence {zn} plex numbers, we
need only