Complex power raised over real number.

1. Aug 22, 2012

PrashntS

1. I actually dont know if such kind of operation is even allowed.

A friend of mine raised this question, that can we raise a complex power over a real number. I solved it this way. Is this correct?

http://i45.tinypic.com/254vwux.jpg
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: Aug 22, 2012
2. Aug 22, 2012

SammyS

Staff Emeritus
Yes, that's fine.

3. Aug 22, 2012

Millennial

Yes, you did it correctly. I am not quite sure approximating the transcandental functions with decimals, but I suppose that's fine.

4. Aug 22, 2012

Ray Vickson

Your method is OK, but if you plan to use the results in further numerical computations, you should keep more digits of accuracy; nowadays, in this computer age, there is no barrier to retaining more digits. For example, it might be better (depending on future uses) to write
$$3^{5i} = e^{i 5 \ln 3 } = \cos(5 \ln 3) + i \sin(5 \ln 3) \doteq 0.7037573 - 0.7104404 i\, .$$

RGV

5. Aug 22, 2012

PrashntS

Yeah this is obviously much better, I was just looking whether it is even possible or not as none of the books I use has such question.
Would be awesome if you could suggest some good book for complex number (pre collage).

6. Aug 23, 2012

Bacle2

Are you also taking into account that there are infinitely-many solutions depending