, and , forOriginally Posted by ThePerfectHacker
Now raise these to the power , and you have multiple values one
of which will probably be what you are expecting.
I posed to myself two problems:
For the second one I did it seems good.
But for the second one
yet it should be one!?!?
I am guessing the exponent for complex numbers is defined for where Thus, my manipulation in problem 2 is out of the domain of the definition and it causes problems.
Am I right?
You have to be careful with complex numbers, not all the properties you know from the real numbes (about square roots, powers, logarithms, ...) still hold for complex numbers.
In particular, the rule no longer holds.
The correct version is .
And this again yields for .
This has to do with the analytical continuation of the natural logarithm as a complex function and the fact that for complex values of z, is a periodic function with period .
Thank you TD!, well answered.Originally Posted by TD!
I knew that all of these "paradoxes" I was getting was the result of incorrect rules for complex numbers. It is like the square root of the product of two negative numbers IS NOT the product of the square roots. I just never studied complex analysis.
Indeed, that's also one of the consequences. It follows from the fact that when you analytically continue the square root function as a complex function, you'll have to introduce a branch cut and this makes it possible for the 'old rule' we had for the real case to give false results in the complex case.Originally Posted by ThePerfectHacker