How to raise a complex number to the power of another complex number?

Solution 1:

First you need to realize that this is a multi-valued function.

Let us choose the principal logarithm for convenience. So our argument $\theta$ will lie between $(-\pi,\pi]$.

We then write $a+ib = r e^{i \theta} = e^{(ln(r) + i \theta)}$.

Now, we get $(a+ib)^{c+id} = e^{(ln(r) + i \theta)(c+id)}$. Do the necessary algebraic manipulations in the exponent to get $e^{(cln(r) - d \theta) + i(d ln(r) + c \theta)}$. You might also want to take a look at the previous question asked on a similar topic.

Solution 2:

Well, assuming principal values of the complex logarithm (otherwise much craziness ensues):

$$(a+bi)^{c+di}=\exp((c+di)\ln(a+bi))$$

$$=\exp((c+di)(\ln|a+bi|+i\arg(a+bi)))$$

$$=\exp((c\ln|a+bi|-d\arg(a+bi))+i(c\arg(a+bi)+d\ln|a+bi|))$$

$$=\exp((c\ln|a+bi|-d\arg(a+bi)))\exp(i(c\arg(a+bi)+d\ln|a+bi|))$$

and I'll let you finish it at this point, using the fact that $\exp(ix)=\cos\;x+i\sin\;x$