# Complex power raised over real number.

• PrashntS
In summary, a friend asked if it is possible to raise a complex power over a real number, and the author found a solution that is accurate to more digits.f

#### 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

## The Attempt at a Solution

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Yes, that's fine.

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

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

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).

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