Afraid to Manipulate Complex Numbers

[nonequilibrium]

Hello. I'm currently following a course in Complex Analysis, but I'm often afraid of manipulating certain expressions. It is well known that certain "intuitively obvious" actions which are true for real numbers are not true for complex numbers, a simple one being $$\sqrt{-1}\sqrt{-1} \neq \sqrt {(-1)(-1)}$$ and many others; there are quite some sites that warn you for these traps, but I can't seem to find any site which then tells me what is allowed. For example, instead of just saying "$$\sqrt{a}\sqrt{b}$$ does not necessarily equal $$\sqrt{ab}$$", I'd also like the site to say "but what stays true is that $$\sqrt{a}\sqrt{b} = \pm \sqrt{ab}$$". For example, something I'm wondering about: I know $$(a^x)^y = a^{xy}$$ is not generally true anymore, but in what cases can I still do it anyway?

 

[Gib Z]

It's hard to memorize what you can and can't do like that, so instead just remember definitions, definitions, definitions! $x^y = \exp \left( y \log x \right)$

$$a^{xy} = \exp \left( xy \log a \right)$$
$$(a^x)^y = \left( \exp \left(x \log a \right) \right)^y = \exp \left( y \log \left( \exp \left(x \log a \right) \right) \right)$$.

So the 2nd matches the first if we address whether $$\log (\exp z) = z$$ where $z = x \log a$. The subtleties of the complex logarithim and how it behaves with the exponential function is described very here: http://en.wikipedia.org/wiki/Complex_logarithm . Using the definitions we can always reduce problems like the one here to one of the basic questions about complex logs, which that article informs you about.