5.1 Fundamentals

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In chapter 2, section 2.7.1, I mentioned briey the importance of in_nite

series, particularly power series, in estimating values of functions. What it

comes down to is that we can easily add, subtract, multiply and except in

the case of zero, divide real numbers, and this is essentially all we can do

with them. The same applies to complex numbers. The only operation that

makes sense otherwise is taking limits, and again this makes sense for complex

numbers also. It follows that if we want to calculate sin(2) or some other

function value, it must be possible to compute the answer, to increasing accuracy,

in terms of some _nite number of repeated additions, multiplications,

subtractions and divisions, or there isn't any meaning to the expression. We

can accept that we may never get an exact answer in a _nite number of operations,

but if we can't get an estimate and know the size of the uncertainty

with a _nite number of standard operations, and if we cannot guarantee that

we can reduce the uncertainty below any amount that is desired by doing

more simple arithmetic, then sin(2) simply doesn't have any meaning. The

same holds for all the other functions. Even the humble square root of 2 exists

only because we have a means of computing it to any desired precision.

And the only way of doing this must involve only additions, subtractions,

multiplications and divisions, because this is all there is. Your calculator or

computer must therefore use some form of truncated in_nite series in order

to compute p2 or arctan 1=4 or whatever. A more expensive calculator may

use more terms, or it may use a smarter series which converges faster, or it

131

132 CHAPTER 5. TAYLOR AND LAURENT SERIES

may do some preprocessing using properties of the function, such as reducing

trig functions by taking a remainder after subtracting o_ multiples of 2_ to

evaluate sin(100). But it must come down to in_nite series except for the

cases where it can be calculated exactly in a _nite number of operations.

It follows that series expansions for functions is absolutely fundamental, and

that the question of when they converge is also crucial. A calculator that

tried to compute something by using the series

1 + 1=x + 1=2x2 + 1=3x3 +_ _ _

would run into trouble at x = 1, but it would produce an answer- one which

is meaningless. Somebody has to design the calculator and that someone has

to know when garbage is going in if garbage is not to come out.

The idea of an in_nite series representations of a function then is simply that

of always being able to add on an extra little bit which will make the result

closer to the 'true' answer, and knowing something about the precision we

have attained at each step. And that is all in_nite series are about.

This comes out in the jargon as:

f(z) =

1 X

1

akzk

or something similar, where we have a way of calculating the ak. And what

this means is that if

Sn(z) =

n

X1

akzk

is the sum of the _rst n terms, the sequence Sn(z) has a limit for every z.

And what this means is that there is for each z some complex number w such

that if you stipulate a precision ", a small positive real number, then there

is some N, a critical number of steps, such that after that many steps, the

partial sum Sn for n > N is always within the desired accuracy of the answer

w. In algebra:

n > N ) jSn 􀀀wj < "

Putting this together, we say that

f(z) =

1 X

1

akzk

, 8" 2 R+; 9N 2 N : n > N ) jf(z)􀀀

n

X1

akzk

j < "

5.1. FUNDAMENTALS 133

This blows the mind at _rst sight, but it only says compactly what the

preceding half page said. Read it as: 'f(z) is expressed as the in_nite sum

P11

akzk means that for any accuracy " desired in the answer, we can always

_nd some number of terms N, such that if we calculate the sum to at least

N terms, we are guaranteed to be within " of the answer.'

Note that this makes as much sense for z complex as it does for z real.

What is essential is that you read such expressions for meaning and don't

simply switch o_ your brain and goggle at it. It shouldn't be necessary to say

this, it should have come with every small bit of Mathematics that you ever

did, but cruel experience has taught me that too many people stop thinking

about meaning and start trying to memorise lines of symbols instead. I have

been to too many Engineering Honours seminars to have any faith in students

having grasped the fundamentals, and without the fundamentals it turns into

ritualistic nonsense rather fast.

From the above de_nition, it should be very clear that if I give you a new

function of a complex variable, I must either tell you how to calculate those

aks, or equivalently I must tell you how to calculate it in terms of other

functions you already know, where you have been given the corresponding

aks.

When you _rst met the cos and sin functions, they were probably de_ned

in terms of the x and y coordinates of a point on the unit circle. If they

weren't, they should have been. This is all very well, but you ought to have

asked how to calculate them. You cannot expect your hand calculator to

work out cos(2) by drawing bloody big circles. At some later stage, you met

the Taylor-MacLaurin series:

sin(x) = x􀀀

x3

3!

+

x5

5! 􀀀_ _ _

and this should have cheered you up somewhat. This is something your

calculator can do. The _rst question that you should be all agog to _nd out

the answer to is, how did we get the series and is it actually right? And

the second question any reasonably suspicious engineer should ask is, does it

always converge? And the third question is, given that it converges and to

the right answer, how many steps does it take to get a reasonably accurate

answer? How many steps do we need to get within 10􀀀4 of the true result,

for example? This last is a severely practical matter: a computer can do

some millions of oating point operations in a second, and TF1 can do about

1012 ops. But the de_nition of convergence only says that an N has to exist

134 CHAPTER 5. TAYLOR AND LAURENT SERIES

for any ", it doesn't say that it has to be some piddling little number like

10100 or less. There must be a function such that when " is 10, N is 101010

.

This means that we would never know the value of f(1) to within an order

of magnitude before the stars turn into black cinders. One would like to do

a little better than that for sin(1).

There are satisfactory answers to these questions for the function sin(x). It

is worth understanding how it was done for sin(x) so you can do the same

thing for other functions, in particular for sin(z) when z is complex. I have

already assured you that there is a power series for sin(z), and you may have

learnt it. But knowing how to get it is rather more useful.

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