Complex Powerfunction: Exploring Properties & Logical Explanations

[giokara]

Hi

I just got an introduction in complex analysis and some things are still not so clear. What troubles me the most is a property of taking the power of a complex variable. We have seen that:
$$(z^a)^b = z^{ab} e^{2ki\pi b}$$
I can prove that formula but I can't understand it. Does this mean that when we take b = 1/a,
$$(z^a)^{\frac{1}{a}} \neq z$$
in the general case?
If that is true (which I assume), is there a logical explanation for it? I see it comes from the branch cut that is inserted to use the definition of a power (z^a = exp(a*log(z))) but I can't see why exactly this results in this strange property for a complex powerfunction..

Thx!

 

[awkward]

Hi

I just got an introduction in complex analysis and some things are still not so clear. What troubles me the most is a property of taking the power of a complex variable. We have seen that:
$$(z^a)^b = z^{ab} e^{2ki\pi b}$$
I can prove that formula but I can't understand it. Does this mean that when we take b = 1/a,
$$(z^a)^{\frac{1}{a}} \neq z$$
in the general case?
If that is true (which I assume), is there a logical explanation for it? I see it comes from the branch cut that is inserted to use the definition of a power (z^a = exp(a*log(z))) but I can't see why exactly this results in this strange property for a complex powerfunction..

Thx!

It's not all that mysterious, even in the reals. Take z = -1, a = 2. What do you get?

The main thing you need to remember about complex powers is that z^a is not uniquely defined for non-integer a. Again this is not all that mysterious. In the reals we define x^(1/2) to be the positive square root, but that is only a convention which serves to make the identities work out nicely. Sometimes the negative square root is the "right answer". For the complex numbers, no convention works-- so you have to learn to live with multiple answers for the square root and other non-integer powers.