Complex power raised over real number.

[PrashntS]

1. I actually don't 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

Homework Statement

Homework Equations

The Attempt at a Solution

 

[SammyS]

Yes, that's fine.

 

[Millennial]

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

 

[Ray Vickson]

1. I actually don't 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

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

 

[PrashntS]

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

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).

 

[Bacle2]

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

on your choice of branch/argument?

 

[PrashntS]

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

on your choice of branch/argument?

well that's obvious, isn't it? trigo fn is periodic, i just stayed in principle branch