2.3 The Square Root: w = z

Back

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The square root function, f(z) = z

1

2 is another function it pays to get a

handle on. It is inverse to the square function, in the sense that if you square

2.3. THE SQUARE ROOT: W = Z

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the square root of a a number you get the number back. This certainly works

for the real numbers, although you may not have a square root if the number

is negative. We have just convinced ourselves (by thinking about carpets)

that every complex number except zero has precisely two square roots. So

how do we get a well de_ned function from C to itself that takes a complex

number to a square root?

In the case of the real numbers, we have that there are precisely two square

roots, one positive and one negative, except when they coincide at zero. The

square root is taken to be the positive one. The situation for the complex

plane is not nearly so neat, and the reason is that as we go around the circle,

looking for square roots, we go continuously from one solution to another.

Start o_ at 1 + i0 and you will surely agree that the obvious value for its

square root is itself. Proceed smoothly around the unit circle. To take a

square root, simply halve the angle you have gone through.

By the time you get back, you have gone through 2_ radians, and the preferred

square root is now 􀀀1 + i0. So whereas the two solutions formed two

branches in the case of the reals, and you could only get from one to the

other by passing through zero, for C there are continuous paths from one

solution to another which can go just about anywhere.

Remember that a function is an input-output machine, and if we input one

value, we want a single value out. We might settle for a vector output in

C _C , but that doesn't work either, because the order won't stay _xed. We

insist that a function should have a single unique output for every input,

because all hell breaks loose if we try to have multiple outputs. Such things

are studied by Mathematicians, who will do anything for a laugh, but it

makes ideas such as continuity and di_erentiability horribly complicated. So

the complications I have outlined to force the square root to be a proper

function are designed to make your life simpler. In the real case, we can

simply choose px and 􀀀px to be two neat functions that do what we want,

at least when x is non-negative. In the complex plane, things are more

complicated.

The solution proposed by Riemann was to say that the square root function

should not be from C to C , but should be de_ned on the Riemann surface

illustrated in _gure 2.13. This is cheating, but it cheats in a constructive

and useful manner, so mathematicians don't complain that Riemann broke

the rules and they won't play with him any more, they rather admire him

48 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

O

Q Q’

P P’

Figure 2.14: The Square function through the Riemann Surface

for pulling such a line2.

If you build yourself a surface for the square function, then you project it

down and squash the two sheets (cones in my picture) together to map it

into C , then you can see that there is a one-one, onto, continuous map from

C to the surface, S, and then there is a projection of S on C which is two-one

(except at the origin). So there is an inverse to the square function, but it

goes from S to C . This is Riemann's idea, and it is generally considered very

cool by the smart money.

I have drawn the pictures again in _gure 2.14; you can see the line in the

lower left copy of C (or a bit of it) where I have glued OP to OQ' and OP'

to OQ, and then both lines got glued together by the projection. The black

arrow going down sends each copy of C to C by what amounts to the identity

map. This is the projection map from S. The black arrow going from right

to left and slightly uphill is the square function onto S. The top half of the

complex plane is mapped by the square function to the top cone of S, and

the bottom half of C is mapped to the lower cone.

2Well, the good mathematicians feel that way. They like style. Bad mathematicians

don't like this sort of thing, but life is hard and unkind to bad mathematicians who spend

a lot of the time feeling stupid and hating themselves for it. We should not add to their

problems.

2.3. THE SQUARE ROOT: W = Z

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The last black arrow going left to right is the square root function, and it

is a perfectly respectable function now, precisely the inverse of the square

function.

So when you write f(z) = z2, you MUST be clear in your own mind whether

you are talking about f : C 􀀀! C or f : C 􀀀! S. The second has an inverse

square root function, and the former does not.

2.3.1 Branch Cuts

Although the square function to the Riemann surface followed by the projection

to C doesn't have a proper inverse, we can do the following: take half a

plane in C , map it to the Riemann surface, remove the boundary of the half

plane, and project it down to C . This has image a whole plane (the angle

has been doubled), with a cut in it where the edge of the plane has been

taken away. For example, if we take the region from 0 to _, but without the

end angles 0 and _, the squaring map sends this to the whole complex plane

with the positive X-axis removed. This map has an inverse, (r; _);(r1=2; _

2)

which pulls it back to the half plane above the X-axis.

Another possibility is to take the half plane with positive real part, and

square that. This gives us a branch cut along the negative real axis. We can

then write

f1(z) = f1(r; _) = (r1=2;

_

2

)

for the inverse, which is called the Principal Square Root. It is called a

branch of the square root function, thus confusing things in a way which is

traditional. We say that this is de_ned for 􀀀_ < _ < _.

Suppose we take the half-plane with strictly negative real part: this also gets

sent to the complex plane with the negative real axis removed. (We have to

think of the angles, _ as being between _=2 and 3_=2.) Now we get a square

root of (r; _) which is the negative of its value for the principal branch. I

shall call this the negative of the Principal branch.

Exercise 2.3.1 Draw the pictures of the before and after squaring, for the

two branches just described. Con_rm that (1; _=4) is the unique square root

of (1; _=2) for the Principal branch, and that (1; 5_=4) is the unique square

root of (1; _=2) for the negative of the Principal branch. Note that (1; _=4) =

􀀀(1; 5_=4).

50 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

Taking branches by choosing any half plane you want is possible, and for

every such branch there is a branch cut, and a unique square root, For every

such branch there is a negative branch obtained by squaring the opposite

half plane, and having the same branch cut. This ensures that in one sense

every complex number has two square roots, and yet forces us to restrict the

domain to ensure that we only get them one at a time.

The point at the origin is called a branch point; I _nd the whole terminology

of 'branches' unhelpful. It suggests rather that the Riemann surface comes

in di_erent lumps and you can go one way or the other, getting to di_erent

parts of the surface. For the Riemann surface associated with squaring and

square rooting, it should be clear that there is no such thing. It certainly

behaves in a rather odd way for those of us who are used to moving in three

dimensions. It is rather like driving up one of those carp parks where you go

upward in a spiral around some central column, only instead of going up to

the top, if you go up twice you discover that, SPUNG! you are back where

you started. Such behaviour in a car park would worry anyone except Dr.

Who. The origin does have something special about it, but it is the only

point that does.

The attempt to choose regions which are restricted in angular extent so

that you can get a one-one map for the squaring function and so choose a

particular square root is harmless, but it seems odd to call the resulting bits

'branches'. (Some books call them 'sheets', which is at least a bit closer to

the picture of them in my mind.)

It is entirely up to you how you choose to do this cutting up of the space into

bits. Of course, once you have taken a region, squared it, con_rmed that the

squaring map is one-one and taken your inverse, you still have to reckon with

the fact that someone else could have taken a di_erent region, squared that,

and got the same set as you did. He would also have a square root, and it

could be di_erent from yours. If it was di_erent, it would be the negative of

yours.

Instead of di_erent 'branches', you could think of there being two 'levels',

corresponding to di_erent levels of the car park, but it is completely up to

you where you start a level, and you can go smoothly from one level to the

next, and anyway levels 1 and 3 are the same.

This must be hard to get clear, because the explanations of it usually strike

me as hopelessly muddled. I hope this one is better. The basic idea is fairly

2.3. THE SQUARE ROOT: W = Z

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easy. Work through it carefully with a pencil and paper and draw lots of

pictures.

2.3.2 Digression: Sliders

Things can and do get more complicated. Contemplate the following question:

Is w = p(z2) the same function as w = z?

The simplest answer is 'well it jolly well ought to be', but if you take z = 1

and square it and then take the square root, there is no particular reason to

insist on taking the positive value. On the other hand, suppose we adopt the

convention that we mean the positive square root for positive real numbers,

in other words, on the positive reals, square root means what it used to mean.

Are we forced to take the negative square root for negative numbers? No,

we can take any one we please. But suppose I apply two rules:

1. For positive real values of z take the (positive) real root

2. If possible, make the function continuous

then there are no longer any choices at all. Because if we take a number such

as ei_ for some small positive _, the square is ei2_ and the only possibilities

for the square root are _ei_, which since r cannot be negative means ei_ or

ei(_+_, and we will have to choose the former value to get continuity when

_ = 0. We can go around the unit circle and at each point we get a unique

result: in particular p(􀀀1)2) = 􀀀1.

I could equally well have chosen the negative value everywhere, but with both

the above conventions, I can say cheerfully that

pz2 = z

If I drop the continuity convention, then I can get a terrible mess, with signs

selected any old way.

The argument for

(pz)2

52 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

is simpler. If you take z and look at its square roots, you are going up from

C to the Riemann surface that is the double level spiral car-park space. you

can go up to either level from any point (except the origin for which there is

only one level). If I square the answer I will get back to my starting point,

whatever it was. So

z = (pz)2

is unambiguously true, although it is expressing the identity function as a

composite of a genuine function and a relation or 'multi-valued' function.

Now look at

w = pz2 + 1

Again the square root will give an ambiguity, but I adopt the same two rules.

So if z = 1, w =p2. At large enough values of z, we have that w is close to

z. The same argument about going around a circle, this time a BIG circle,

gives us a unique answer. 100i will have to go to about 100i and 􀀀100 will

have to go to about 􀀀100.

It is by no means clear however that we can make the function continuous

closer in to the origin. f(0) = 1 would seem to be forced if we approach zero

from the right, but if we approach it from the left, we ought to get 􀀀1. So

the two rules given above cannot both hold. Likewise, _i both get sent to

zero; If we have continuity far enough out, then we can send 10i to i times

the positive value of p99. But what do we do for 0:5i? Do we send it to

p3=4 or􀀀p3=4? Or do we just shrug our shoulders and say it ismultivalued

hereabouts?

If we just chop out the part of the imaginary axis between i and 􀀀i, we have

a perfectly respectable map which is continuous, and sends i(1 + _) into i_

when _ > 0, sends 􀀀1 to 􀀀

p2, 1 to +p2. It has image the whole complex

plane except for the part of the real axis between 􀀀1 and 1. Call this map

f. You can visualise it quite clearly as pulling the real axis apart at the

origin, with points close to zero on the right getting sent (almost) to 1 and

points close to zero on the left getting sent (almost) to 􀀀1. The two points

_i get sucked in towards zero. Because of the slit in the plane, this is now a

continuous map, although we haven't de_ned it on the points we threw out.

There is also a perfectly respectable map 􀀀f which sends z to 􀀀f(z). This

has exactly the same domain and range space, C with a vertical slit in it,

between 􀀀i and i, and it has the same range space, C with a horizontal slit

in it, between 􀀀1 and 1. It is just f followed by a rotation by 180o.

2.3. THE SQUARE ROOT: W = Z

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We now ask for a description of the Riemann surface for pz2 + 1. You might

think that asking about Riemann surfaces is an idle question prompted by

nothing more than a desire to draw complicated surfaces, but it turns out

to be important and very practical to try to construct these surfaces. The

main reason is that we shall want to be able to integrate along curves in due

course, and we don't want the curve torn apart.

The Riemann surface associated with the square and square root function

was a surface which we pictured as sitting over the domain of the square root

function, C , and which projected down to it. Then we split the squaring

map up so that it was made up of another map into the Riemann surface

followed by the projection. Actually, the Riemann surface is just the graph

of w = z2, but instead of trying to picture it in four dimensions, we put it

in three dimensions and tried not to think about the self-intersection this

caused.

We could construct the above surface as follows: _rst think of the square

root function. Take a sector of the plane, say the positive real axis and the

angle between 0 and _=2. Now move it vertically up above the base plane.

I choose one particular square root for the points in this sector, say I start

with the ordinary real square root on the positive real line. This determines

uniquely the value of the square root on the sector, since w = z2 is one-one

here, so the square root is just half the sector. I can do the same for another

sector on which the square is still one-one, say the part where the imaginary

component is positive. This will overlap the quarter plane I already have;

I make sure that everything agrees with the values on the overlap. I keep

going, but when I get back to the positive real axis, I discover that I have

changed the value of the square root, so instead of joining the points, I lift

the new edge up. I keep going around, and now I get di_erent answers from

before, but I can continue gluing bits together on the overlap. When I have

gone around twice, I discover that the top edge now really ought to be joined

up to the starting edge. So I do my Dr. Who act and identify the two edges.

The other way to look at it is to take two copies of the complex plane, and

glue them together as in _gure 2.14. We know there are two because of the

square root, and we know that they are joined at the origin because there is

only one square root of zero. We clearly pass from one plane to another at a

branch cut, which can be anywhere, and then we go back again a full circuit

later.

Now I shall do the same thing with pz2 + 1. But before tackling this case,

54 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

a short digression.

Example 2.3.1 In the television series 'Sliders', the hero generated a disk

shaped region which identi_ed two di_erent universes. Suppose there are two

people intending to slide into a new universe and they see this disk opening

into a tunnel in front of them. One of them walks around the back of the

disk. If this one sees the other side of the disk and steps through it, and if

the other person goes through the other side of the disk at the same time, is

it true that they must come out in the same place? Do they bump into each

other?

Solution

It is probably easiest to think of this a dimension lower down. Take two sheets

of paper, two universes. Draw a line segment on each. This is the 'door into

Summer', the Stargate.

What we do is to identify the one edge of the line segment in one universe with

the opposite edge in the other universe. To make this precise, take universe

A to be the plane (x; y; 1) for any pair of numbers x; y, and universe B to be

the set of points (x; y; 0). I shall make my 'gateway' the interval (0; y; n) for

􀀀1 _ y _ 1, for both n = 0;1.

Now I _rst cut out the interval of points in the 'stargate',

(0; y; n); 􀀀1 _ y _ 1; n = 1;2

I do this in both universes.

Now I pull the two edges of the slits apart a little bit. Then I put new boundaries

on, one on each side. I have doubled up on points on the edge now,

so there are two origins, a little way apart, in each universe. I call them

0 and 00 respectively, so I have duplicate points (0; y; n) and (00; y; n) for

􀀀1 _ y _ 1; n = 1;2. A crude sketch is shown in _gure 2.15. Now I glue

the left hand edge of the slit in one universe to the right hand edge of the slit

in the other universe, and vice versa. So

(0; y; 0) = (00; y; 1) & (0; y; 1) = (00; y; 0) 􀀀 1 _ y _ 1

This will make the path shown by the dotted line in _gure 2.16 continuous.

I joined up the opposite two sides of the cut in each universe in the same

way, but I don't have to. One thing I can do is to have another universe,

2.3. THE SQUARE ROOT: W = Z

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0 0

Figure 2.15: The construction of the Stargate

World A

World B

Figure 2.16: The Dotted line is a continuous path in the twin universes

56 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

and join to that one. So if two of the slider gang go through the gate on

opposite sides, they could emerge in the same universe on opposite sides of

the connecting disk, or in two di_erent universes. They won't bump into each

other, they will be on opposite sides of the disk, but they may or may not be

in the same universe.

On the other hand, nipping back smartly where you have just come from,

walking around the disk, and then going in the same side would get them

together again.

On this model.

If someone ever does invent a gateway for travelling between universes, the

mathematicians are ready for talking about it3.

The reason for thinking about multidimensional car parks, sliders and bizarre

topologies, is that it has everything to do with the Riemann surface for

w = f(z) = pz2 + 1

We need to take both f and 􀀀f, and we have quite a large region in which

we can have each branch of f single valued and 1-1, namely the whole plane

with the slit from i to 􀀀i removed. So we have two copies of C with slits in

them.

We also have two similar looking copies of slitted C s (but with horizontal

slits) ready for the image of the new map.

We have to join the two copies of C across the slits. This is exactly what our

picture of the two dimensional inter-universe sliders was doing.

In this case, we can label our two universes as f and 􀀀f. This is going to

tell us what we are going to actually do with the linked pair of universes.

Points on the left hand side of the slit for universe f are de_ned to be close to

the points on the right hand side of the slit for universe 􀀀f and vice versa. So

a path along the real axis from 􀀀1 towards 1 in the universe f slips smoothly

into the universe 􀀀f at the origin. You can retrace your path exactly. If you

start o_ in Universe f at +1 going left, then you slide over into universe 􀀀f

3Actually they've been ready for well over a century. Riemann discussed this sort of

thing in 1851. It took a while to get down to the level of popular television.

2.3. THE SQUARE ROOT: W = Z

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at the origin. So in this case, it doesn't matter which way you go into the

'gate', you wind up in the same universe- there are only two. If you are a

long way from the gate in either universe, you don't get to _nd out about

the other universe at all. Continuous paths which don't go through the gate

have to stay in the same universe.

Exercise 2.3.2 Construct a complex function needing three 'universes' for

the construction of its Riemann surface.

To see that this is the Riemann surface, observe that if we travel in any path

on the surface, the value of pz2 + 1 varies continuously along the path.

Exercise 2.3.3 Choose a path in the Riemann surface and con_rm that the

value of pz2 + 1 varies continuously along the path. Do this for a few paths,

some passing through the 'gate' described above.

Exercise 2.3.4 Describe the surface associated with the inverse function.

Show that there is a one-one continuous map going in both directions between

the two surfaces.

It is worth pointing out that the Riemann surface can be constructed in

several ways: there is nothing unique about the choice of branch cuts, for

example. It is not so obvious that the Riemann surface is unique in the sense

that there is always a way of deforming one into another. You don't have

the background to go into this, so I shan't. But the text books often give the

impression that branch cuts come automatically with the problem, whereas

they are much less clear cut4 than that.

It is clear that the z2 cut along the positive real axis can be replaced by any

ray from the origin. It might seem however that the slit between i and 􀀀i is

forced. This isn't so, but the proper investigation of these matters is quite

di_cult.

I have avoided de_ning Riemann surfaces, and simply considered them in

rather special cases, because it needs some powerful ideas from Topology to

do the job properly. This seems to be traditional in Complex Analysis, and it

4Aaaagh!

58 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

leaves students rather puzzled as to how to handle them in new cases. There

isn't time in this course to do more than introduce them, but I hope you

can see two things: _rst that quite simple real functions generalise to rather

complicated complex functions, and second that the investigation of them is

full of ideas that take you outside the universe you are used to. The fact that

this is actually useful is one you will have to take for granted for a while.

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