2.8 Other powers

Back

I can now de_ne zw for complex numbers z and w by

zw = exp(w log(z))

which is 'multi-valued', i.e. not a function but an in_nite family of them.

Taking the Principal Branch makes this a function. This agrees with the

ordinary de_nition when w is an integer.

Exercise 2.8.1 Prove that last remark. Does it work for w any real number?

Exercise 2.8.2 Calculate 􀀀1i.

Since we can do in C anything we can do in R of an algebraic sort, we can

_nd more exotic powers. The following exercise should be done in your head

while walking to prove that you know your way around:

Exercise 2.8.3 Calculate ii.

The next one can also be done internally if your concentration is in good

nick:

Exercise 2.8.4 Calculate ( 1􀀀i

p2

)2i

This is good, clean fun. I have tried watching television and doing these

sorts of calculations, and in my view the sums are more fun, although they

may keep you awake at nights. You may be able to see why Gauss and

Euler, two of the brightest men who ever lived, spent some time playing with

the complex numbers a long, long time before they were really much use

for anything. It's just nice to know that something like the square root of

negative one raised to the power of itself is a perfectly respectable number.

Actually a lot of perfectly respectable numbers. Find them all. One of them

is a smidgin over 0:2.

82 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

Навигация