Complex Power of Complex Numbers

How to find the complex power of a complex number? 

Is this possible???

Yes. In this topic, we will find the complex power of a complex number in complex analysis. It is a very interesting and easy method to solve complex power. Here, we use simple math methods to solve complex power of a complex number.

Complex Power of Complex Numbers

General powers of a complex number `z = x + yi` are defined by formula:

`z^c = e^(c lnz)`            (c complex, z `\ne` 0)

Since lnz is infinitely many-valued, `z^c` will in general, be multivalued. The particular value

                                                          `z^c = e^(c Lnz)` 

is called the principal value of `z^c`.

If c = 1,2,..., then `z^n` is single valued and identical with the usual nth power of z.

If c = -1, -2,...., the situation is similar.

If c = 1/n, where n = 2,3,..., then

                                                 `z^c = e^((1/n)lnz)`          (`z\ne0`)