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matrix of partial derivatives
u u
x y
v v
x y
If you learnt nothing else from second year Mathematics you may still be
able to hold your head up high if you grasped the idea that the above matrix
is the two dimensional version of the slope of the tangent line in dimension
one. It gives the linear part corresponding to the slope of the a ne map
which best approximates f at each point.
If f R R is a di erentiable function then df dx at any value of t is some
real number m. Well what we really mean is that the map y mx f t mt
is the a ne map which is the best approximation to f at t. It has slope m
and the constants have been xed up to ensure that it passes through the
point t f t .
This is the old diagram from school days gure 3.1.
In a precisely parallel way the matrix of partial derivatives gives the linear
part of the best a ne approximation to the map f R2 R2 . But at
89
90 CHAPTER 3. C DIFFERENTIABLE FUNCTIONS
y=mx+f(t)-mt
y=f(x)
f(t)
t
dy
= m
dx t
Figure 3.1 The Best A ne Approximation to a real di erentiable function
any point x iy if f is di erentiable in the complex sense this must be just
a linear complex map i.e. it multiplies by some complex number. So the
matrix must be in our set of complex numbers. In other words for every
value of x it looks like
a b
b a
for some real numbers a b which change with x.
This forces us to have the famous Cauchy Riemann equations
u x v y and u y v x
It is important to understand what they are saying there are plenty of maps
from R2 to R2 which are real di erentiable and will have the matrix of partial
derivatives not satisfying the CR conditions. But these will not correspond
to being a linear approximation in the sense of complex numbers. There
is no complex derivative in this case. For the complex derivative to exist
in strict analogy with the real case the matrix must be antisymmetric and
have the top left and bottom right values equal. This is a very considerable
restriction and means that many real di erentiable functions will fail to be
complex di erentiable.
3.1. TWO SORTS OF DIFFERENTIABILITY 91
Exercise 3.1.1 Let denote the conjugation map which takes z to z. This
is a very di erentiable map from R2 to R2 . Write down its derivative matrix.
Is conjugation complex di erentiable anywhere
On the other hand the de nition of the derivative for a real function such
as f x x2 in the real case was
dy f t f t
j lim
t
dx
We know that at t 1 and f x x2 we have
dy 1 2 12
j lim
1
dx
and of course
1 2 12
lim
2 2
lim
lim 2
2
Now all this makes sense in the complex numbers. So if we want the derivative
of f z z2 at 1 i we have
1 i 2 1 i 2
f 1 i lim
1 i 2 2 2 1 i 1 i 2
lim
lim 2 1 i
2 1 i
Here is some complex number but this has no e ect on the argument. By
going through the above reasoning with z in place of 1 i you can see that
the derivative of f z z2 is 2z regardless of whether z is real or complex.
92 CHAPTER 3. C DIFFERENTIABLE FUNCTIONS
If we write the function f z z2 as
x iy u iv x2 y2 i 2xy
we see that u x v y 2x and u y v x so the CR equations
are satis ed. And the derivative is
2x 2y
2y 2x
as a matrix and hence 2x i2y as a complex number. So everything ts
together neatly.
Moreover the same argument holds for all polynomial functions. The argu
ments to show the rules for the derivative of sums di erences products and
quotients all still work. You can either go back dig in your memories and
check or take my word for it if you are the naturally credulous sort that
school teachers and con men approve so heartily.
It might be worth pointing out that the reason Mathematicians like abstrac
tion and talk of doing vector spaces over arbitrary elds for instance is that
they are lazy. If you do it once and nd out exactly what properties your
arguments depend upon you won t have to go over it all again a little later
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 [ Pobierz całość w formacie PDF ]