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102 CHAPTER 3. C DIFFERENTIABLE FUNCTIONS
Of course we could have got the same answer by standard methods but this
is rather neat.
We shall discover later that the curves x iy C the equipotentials
decompose the plane into a family of curves for various values of C which
are orthogonal to the curves x iy D for various values of D. This
means that we can look upon the latter curves as the streamlines of the ow.
It should be obvious to you for physical reasons that the ow should always
be orthogonal to the curves of constant potential. If it isn t obvious ask.
In other words the solutions to the vector eld regarded as a system of ODEs
can be obtained directly from integrating a complex function. Thinking
about this leads to the conclusion that this is not too surprising but again
it is rather neat.
3.3 Conformal Maps
There was an exercise in chapter two which invited you to notice that if
you took any of the functions you had been working with at the time all of
which were analytic almost everywhere then the image by such a function of
a rectangle gave something which had corners. Moreover although the edges
of the rectangle were sent to curves the curves intersected at right angles.
The only exception was the case when f z z2 and the corner was at the
origin.
The question was asked why is this happening and why is there an exception
in the one case
If you are really smart you will have seen the answer if you take a corner
where the edges are lines intersecting at right angles then if the map f is an
alytic at the corner it may be approximated by its derivative there. And this
means that in a su ciently small neighbourhood the map is approximable
as an a ne map multiplication by a complex number together with a shift.
And multiplication by a complex number is just a rotation and a similarity.
None of these will stop a right angle being a right angle. The only exception
is when the derivative is zero when all bets are o .
It is clear that not just right angles are preserved by analytic functions
any angle is preserved. This is rather a striking restriction forced by the
3.3. CONFORMAL MAPS 103
properties of complex numbers and derivatives.
This property of a complex function is called isogonality2 or conformality
with the latter sometimes being restricted to the case where the sense of the
angle is preserved. For our purposes the term conformal means that angles
are preserved everywhere which is guaranteed if the map is analytic and has
derivative non zero everywhere.
Exercise 3.3.1 For which complex numbers w is multiplication by w going
to preserve the sense of two intersecting lines
Exercise 3.3.2 Give an example of a conformal map in this sense which is
not analytic.
There are a lot of applications of Complex Function Theory which depend
on this property I do not alas have time to do more than warn you of what
your lecturers in Engineering may exploit at some later time.
It is very commonly desired to transform some one shape in the plane into
some other shape by a conformal map. Some very remarkable such trans
forms are known see 11 for a dictionary of very unlikely looking conformal
maps. See 9 for the Schwartz Christo el transformations which take the
half plane to any polygon and are conformal on the interior.
It is a remarkable fact that
Theorem 3.1 The Riemann Mapping Theorem
If U is some connected and simply connected region of the complex plane i.e.
it is in one piece and has no holes in it and if it is open i.e. every point in
the U has a disk centred on it also contained in U then providing U is not
the whole plane there is a 1 1 conformal mapping of U onto the interior of
the unit disk. 2
3
2
From the greek isos meaning equal and agon an angle as in pentagon and polygon.
3
Malcolm suggested that I point out that the selection of the interior of the unit disk is
for ease of stating the theorem. It works for a much larger range of regions it is particularly
useful on occasion to take a half plane as the universal region onto which all manner of
unlikely regions can be taken by conformal maps.
104 CHAPTER 3. C DIFFERENTIABLE FUNCTIONS
It follows that for any two open regions of C which are connected and simply
connected there is an invertible conformal map which takes one to the other.
This may seem somewhat unlikely but it has been proved. See 10 for
details.
Chapter 4
Integration
4.1 Discussion
Since we have discussed di erentiating complex functions it is now natural
to turn to the problem of integrating them.
Brooding on what it might mean to integrate a function f C C we
might conclude that there are two factors which need to be considered.
The rst is that integration ought to still be a one sided inverse to di er
entiation di erentiating an inde nite integral of a complex function should
yield the function back again. The second is that integration ought still to
be something to do with adding up numbers associated with little boxes and
taking limits as the boxes get smaller.
We have just been discussing writing out a vector eld as the conjugate of a
complex function so there is a good prospect that we can integrate complex
functions over curves by thinking of them as vector elds. In second year you
managed to make sense of integrating vector elds over curves and surfaces
and should now feel cheerful about doing this in the plane. So your experience
of integration already extends to two and three dimensions and you recall
I hope the planar form of the Fundamental Theorem of Calculus known as
Green s Theorem. If you don t look it up in your notes you re going to need
it.
On the other hand we could just take the real and imaginary parts separately
105
106 CHAPTER 4. INTEGRATION
and integrate each of these in the usual way as a function of two variables.
This would give us some sort of complex number associated with a function
and a region in C. If we were to try to integrate the function 2z in this way
to get an inde nite integral we would get x2y iy2x which is not complex
di erentiable except at the origin. If the FTC is to hold di erentiating an
inde nite integral ought to get us back to the thing integrated and here it
does no such thing. So we conclude that this is not a particularly useful way
to de ne a complex integral.
Now the derivative of a complex function is a complex function so the integral
of a complex function should also be a complex function. So integrating
functions from C to C to get other functions from C to C must be more
like integrating functions from R to R than integrating or vector elds. This
leads to the issue what do we integrate over If we integrate over regions
in C then any version of the Fundamental Theorem of Calculus has to be
some variant of Green s Theorem and must be concerned with relating the
integral over the region of one function with the integral over the boundary
of another. So we seem to need to integrate complex functions over curves if
we need to integrate them over regions. And we knowhowto integrate along
curves because a complex function f z is a vector eld in an obvious way.
Another argument for thinking that curves are the things to integrate com
plex functions over is that if we have an expression like
Z
f z dz
then the dz ought surely to be dx i dy and this is an in nitesimal complex [ Pobierz całość w formacie PDF ]