Chapter 2
Differentiation
Mathematical study and research are very suggestive of mountaineering. Whymper made several
efforts before he climbed the Matterhorn in the 1860’s and even then it cost the life of four of
his party. Now, however, any tourist can be hauled up for a small cost, and perhaps does not
appreciate the difficulty of the original ascent.
So in mathematics, it may be found hard to
realise the great initial difficulty of making a little step which now seems so natural and obvious,
and it may not be surprising if such a step has been found and lost again.
Louis Joel Mordell (1888–1972)
2.1
First Steps
A
(complex) function
f
is a mapping from a subset
G
⊆
C
to
C
(in this situation we will write
f
:
G
→
C
and call
G
the
domain
of
f
).
This means that each element
z
∈
G
gets mapped to
exactly one complex number, called the
image
of
z
and usually denoted by
f
(
z
). So far there is
nothing that makes complex functions any more special than, say, functions from
R
m
to
R
n
. In
fact, we can construct many familiar looking functions from the standard calculus repertoire, such
as
f
(
z
) =
z
(the
identity map
),
f
(
z
) = 2
z
+
i
,
f
(
z
) =
z
3
, or
f
(
z
) =
1
z
. The former three could be
defined on all of
C
, whereas for the latter we have to exclude the origin
z
= 0. On the other hand,
we could construct some functions which make use of a certain representation of
z
, for example,
f
(
x, y
) =
x

2
iy
,
f
(
x, y
) =
y
2

ix
, or
f
(
r, φ
) = 2
re
i
(
φ
+
π
)
.
Maybe
the
fundamental principle of analysis is that of a limit. The philosophy of the following
definition is not restricted to complex functions, but for sake of simplicity we only state it for those
functions.
Definition 2.1.
Suppose
f
is a complex function with domain
G
and
z
0
is an accumulation point
of
G
. Suppose there is a complex number
w
0
such that for every
>
0, we can find
δ >
0 so that
for all
z
∈
G
satisfying 0
<

z

z
0

< δ
we have

f
(
z
)

w
0

<
. Then
w
0
is the
limit
of
f
as
z
approaches
z
0
, in short
lim
z
→
z
0
f
(
z
) =
w
0
.
This definition is the same as is found in most calculus texts. The reason we require that
z
0
is
an accumulation point of the domain is just that we need to be sure that there are points
z
of the
domain which are arbitrarily close to
z
0
. Just as in the real case, the definition does not require
13
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CHAPTER 2.
DIFFERENTIATION
14
that
z
0
is in the domain of
f
and, if
z
0
is in the domain of
f
, the definition explicitly ignores the
value of
f
(
z
0
). That is why we require 0
<

z

z
0

.
Just as in the real case the limit
w
0
is unique if it exists. It is often useful to investigate limits
by restricting the way the point
z
“approaches”
z
0
.
The following is a easy consequence of the
definition.
Lemma 2.2.
Suppose
lim
z
→
z
0
f
(
z
)
exists and has the value
w
0
, as above. Suppose
G
0
⊆
G
, and
suppose
z
0
is an accumulation point of
G
0
. If
f
0
is the restriction of
f
to
G
0
then
lim
z
→
z
0
f
0
(
z
)
exists and has the value
w
0
.
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 Spring '10
 MARCHESI
 Math, Calculus, Complex number, lim g

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