3.1 Two sorts of Di_erentiability

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Suppose f : C 􀀀! C is a function, taking x + iy to u + iv. We know that if

it is di_erentiable regarded as a map from R2 to R2 , then the derivative is a

matrix of partial derivatives:

_ @u

@x

@u

@y

@v

@x

@v

@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

t

f(t)

dy

dx t = m

y=mx+f(t)-mt

y=f(x)

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

dxjt = lim

_!0

f(t+_)􀀀f(t)

_

We know that at t = 1 and f(x) = x2 we have

dy

dxj1 = lim

_!0

(1+_)2 􀀀12

_

and of course

lim

_!0

(1+_)2 􀀀12

_

= lim

_!0

2_+_2

_

= lim

_!0

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

f0(1 + i) = lim

_!0

(1 + i+_)2 􀀀(1 + i)2

_

= lim

_!0

(1 + i)2 +_2 + 2_(1 + i) 􀀀 (1 + i)2

_

= lim

_!0

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 arguments

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 abstraction,

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 unnecessary

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

lim

z!i

z4 􀀀 1

z 􀀀 i

1It 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

lim

z!i

4z3

1

= 4i3 = 􀀀4i

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.

Example 3.1.2 Is f(z) = jzj2 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

lim

_!0

f(_) 􀀀 f(0)

_

we get

lim

x+iy!0

x2 + y2

x + iy

= lim

x+iy!0

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.

The function f(z) = jzj2 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

_!w

f(_) = z

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

f0(z) =

@u

@x

+ i

@u

@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 expression

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

r

@v

@_

=

@u

@r

;

1

r

@u

@_

= 􀀀

@v

@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 regarded

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 differentiable

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 expansions

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 functions

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.

The following results are mostly obvious or easy to prove and are exact

analogues of the real case:

Proposition 3.1.1 If f and g are functions analytic on a domain E, (i.e.

analytic at every point of E) then

1. f+g is analytic on E

2. f-g is analytic on E

3. wf is analytic on E for any complex or real number w

4. fg is analytic on E

3.2. HARMONIC FUNCTIONS 97

5. f/g is analytic on E except at the zeros of g

2

Proposition 3.1.2 If f; g : C 􀀀! C are analytic functions, then the composite

function f _ g : C 􀀀! C is analytic. 2

If f or g have point singularities but are otherwise analytic, then the composite

is analytic except at the obvious singularities. These results will be used

extensively, and because analytic functions have some remarkable properties

they need to be absorbed.

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