The function w=a^z where a is real and w and z are complex, is a multi-function.

That is, for any value of z there is an infinite number of values of w=a^z.

So if we change the meaning of = to mean "the set of" then we can say w=a^z.

We can see this kind of behaviour without going to complex functions:

y=x^(1/2) where x and y are real, is a multifunction. If we choose x=4 then y=+2 or -2, so y has two values for every x and we can't say +2=-2, but we can say y=x^(1/2) if we understand = to mean "the set of". For every value of x there are two values of y and for every value of y there is one value of x.

Here's an example of a similar paradox:

2=4^(1/2)=4^(1/2)=(4 x 1)^(1/2)=4^(1/2) x 1^(1/2)=2 x -1=-2

In the case w=a^z where w, a, z are all complex also results in w being a multivalued function and as long as we realise this and use = to mean "the set of" then we are OK again.

It's true that if we try and do ordinary arithmetic with them, then we will run into difficulty. We should use set theory or define them on the Riemann sphere.

The apparently strange results shown in your link:

http://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identitiesare due entirely to this multivaluedness.

Don't forget to have a look at the other solutions to this exercise. The exercises in the forum default to date/time order. You need to sort them by subject to see the previous solutions or look at the sorted solutions

https://sites.google.com/site/vascoprat/rtr-solutions/chapter-5Note: The Latex problem is something we know about and doesn't look like being solved any time soon. The best solution is to produce a pdf file from your Latex and attach the pdf file to your post.