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when you come to a new case. I have just done exactly that bit of unnec
essary repetition with my investigation of the derivative of z2 but had you
been prepared to buy the abstraction we could have worked over arbitrary
elds in rst year and you would have known exactly what properties were
needed to get these results. The belief that Mathematicians particularly
Pure Mathematicians are impractical dreamers is held only by those too
dumb to grasp the practicality of not wasting your time repeating the same
idea in new words1 .
Virtually everything that works for R also works for C then. This includes
such tricks as L Hopital s rule for nding limits
Example 3.1.1 Find
z4 1
lim
z i
z i
1
It is quite common for stupid people to claim that they have oodles of common sense
or practicality . My father assured me that I was much less practical and sensible than he
was when he found he couldn t do my Maths homework. I believed him until one day in
my teens I found he had xed a blown fuse by replacing it with a six inch nail. I concluded
that if this was common sense I d rather have the uncommon sort.
3.1. TWO SORTS OF DIFFERENTIABILITY 93
Solution
If z i we get the indeterminate form 0 0 so we take the derivative of both
numerator and denominator to get
4z3
lim 4i3 4i
z i
1
which we can con rm by putting z4 1 z i z i z2 1 .
The Cauchy Riemann equations are necessary for a function to be complex
di erentiable but they are not su cient. As with the case of R di erentiable
maps we need the partial derivatives to be continuous and for complex
di erentiability they must also be continuous and satisfy the CR conditions.
2
Example 3.1.2 Is f z j zj di erentiable anywhere
Solution
The R derivative is the matrix
2x 0
0 2y
This cannot satisfy the CR conditions except at the origin. So f is not
di erentiable except possibly at the origin. If it were di erentiable at the
origin it would have to be with derivative the zero matrix. Taking
f f 0
lim
we get
x2 y2
lim
x iy 0
x iy
lim x iy
x iy 0
Since if x iy is getting closer to zero so is its conjugate. Hence f has a
derivative zero at the origin but nowhere else.
2
The function f z j zj is of course a very nice real valued function which
is to say it has zero imaginary part regarded as a complex function. And as
94 CHAPTER 3. C DIFFERENTIABLE FUNCTIONS
a complex function it fails to be di erentiable except at a single point. As
a map f R2 R2 it has u x y x2 y2 and v x y 0 both of which
are as di erentiable as you can get. This should persuade you that complex
di erentiability is something altogether more than real di erentiability.
What does it mean to have an expression like
lim f z
w
over the complex numbers That is are there any new problems associated
with z and w being points in the plane The only issue is that of the
direction in which we approach the critical point w. In one dimension we
have the same issue the limit from the left and the limit from the right can
be di erent in which case we say that the limit does not exist. Similarly if
the limit as w depends on which way we choose to home in on w we
say that there is no limit. In particular problems coming in to zero down
the Y axis can give a di erent answer from coming in along the X axis or
along the line y x. There are some very bizarre functions few of which
arise in real life but you need to know that the functions you are familiar
with are not the only ones there are. You have led sheltered lives.
In the case where the CR equations for some function f C C are
satis ed and the partial derivatives not only exist but are continuous we
have that the complex derivative of f exists and is given by
u u
f z i
x y
in classical form.
There is a polar form of the CR equations. It is fairly easy to work it out I
give it as a pair of exercises
Exercise 3.1.2 By writing
u r u x x r u y y r
And similarly for u v r and v Show the CR equations require
v r u r u r v r
Exercise 3.1.3 Verify that x sin r derive the corresponding ex
pression for y and deduce that
u x i v x cos i sin u r i v r
3.1. TWO SORTS OF DIFFERENTIABILITY 95
which is the partial derivative in polars.
Exercise 3.1.4 Find the other form of the derivative in polars involving
instead of r in the partial derivatives.
Exercise 3.1.5 We can argue that the formulae
v r u r u r v r
are obvious by writing x r and y r on the basis that r are
just rotated versions of any coordinate frame locally and regarding v and
u as in nitesimals obtained by taking in nitesimal independent increments
r and r . Perhaps for this reason it is common to write the polar form as
1 v u 1 u v
r r r r
This is the sort of reasoning that Euler or Gauss would have thought useful
and gives some Pure Mathematicians the screaming ab dabs. It can be re
garded as a convenient heuristic for remembering the polar form or it can be
regarded as showing that in nitesimals ought to have a place in Mathematics
because they work. Although to be fair to Pure Mathematicians second rate
sloppy thinking with in nitesimals can lead to total garbage. For example if
you had tried to put x r and y r you would have got the wrong
answer. Can you see why this is not a good idea
It is possible as we have seen to have a function which is complex dif
ferentiable at only one point This is rather a bizarre case. Functions like
f z z2 are di erentiable everywhere. If a function f is di erentiable at
every point in an open ball centred on some point z0 then it is a particularly
well behaved function at that point
De nition 3.1.1 If f C C is complex di erentiable at every point
in a ball centred on z0 we say that f is analytic or holomorphic at z0.
De nition 3.1.2 A function f C C is said to be entire if it is analytic
at every point of C.
96 CHAPTER 3. C DIFFERENTIABLE FUNCTIONS
De nition 3.1.3 A function f C C is said to have a singularity at z1
if it is not analytic at this point. This includes the case when it is not de ned
there.
De nition 3.1.4 A function f C C is said to be meromorphic if it
is analytic on its domain and this domain is C except for a discrete set of
singular points.
There is a somewhat tighter de nition of meromorphic given in many texts
which I shall come to later.
I hate to load you down with jargon but this is long standing terminology
and you need to know it so that you don t panic when it is sprung on you
in later years. Very often the singularities of a complex function tell you an
awful lot about it and they come up in Engineering and Physics repeatedly.
There is another de nition of the term analytic which makes sense for real
valued functions and is concerned with them agreeing with their Taylor ex
pansions at every point. The two de nitions are in fact very closely related
but this is a little too advanced for me to get into here. I mention it in
case you have come across the other de nition and are confused. The term
complex analytic is sometimes used for the form I have given. Some authors
insist on using holomorphic until they have shown that holomorphic func
tions are in fact analytic in the sense of agreeing with their Taylor expansion
a Theorem of some importance . Then the theorem states that holomorphic
complex functions are analytic.
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