I thought is would be fun to try a problem in which I had a complex number elevated to a complex power. To do this, I first tried to manipulate the general equation ## z^{w} ## (where ##z ## and ##w## are complex numbers) to look a bit more approachable. My work is as follows:

##z^{w}##

##z^{Re(w)+i Im(w)} ##

##z^{Re(w)}*e^{i Im(w) \ln(z)} ##

##\ln(z) = \ln|z| + \theta i ##

## z^{Re(w)}*e^{(\ln|z|+\theta i) i Im(w)} ##

## z^{Re(w)}*e^{i Im(w) \ln|z|- \theta Im(w)} ##

##z^{Re(w)}*e^{i Im(w) \ln|z|}*e^{-\theta Im(w)} ##

##z^{Re(w)} \frac{|z|^{i Im(w)}}{e^{\theta Im(w)}} ##

So

## z^{w} = z^{Re(w)} \frac{|z|^{i Im(w)}}{e^{\theta Im(w)}} ##

I now decided to test this equation out. I chose the number ## 1+ i ## and elevated it to the power of itself. Using wolframalpha I got that ## (1+i)^{(1+i)} \approx 0.273+ i0.584##

https://www.wolframalpha.com/input/?i=(1+i)^(1+i)

I then used the equation I ended up with and used the same number. In wolframalpha, ## (1+i)^{1} * \frac{\sqrt{2}^{i}}{e^{\frac{\pi}{2}}} \approx 0.125 + i0.266##.

https://www.wolframalpha.com/input/?i=(1+i)^1+((sqrt(2)^(i))/(e^(pi/2)))

I don't thing the values computed are even close enough to say that both expressions are equal so I must've messed up somewhere along the way. It would be very appreciated if someone could point out my mistake. In addition, are there any ways to elevate numbers to complex powers in a 'easier' way than just working from ##z^{w} ##? Thanks.