I am trying to solve two complex equations, but as I've never really dealt with complex numbers and equations before, I've run into trouble.

The equations are as follows:

$\displaystyle T_{L} = i \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \cdot e^{i\phi} \right]$

$\displaystyle J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ iP\xi_{0} - \xi_{0}^{'}\right]$

For the first one:

$\displaystyle e^{i\phi} = - \sin(\phi) + i\cos(\phi)$

;thus:

$\displaystyle T_{L} = - \frac{2\pi r \cdot \sin(\phi)}{nd} + \frac{2\pi r \cdot i\cos(\phi)}{nd}$

;which becomes:

$\displaystyle T_{L} = \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \right]^{2}$

;and through a wild goose chase the second one simply becomes:

$\displaystyle J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ - (P\xi_{0})^{2} - \xi_{0}^{'}\right]$

I honestly do not know what I am doing, a friend of mine gave me some "hints" but he doesn't really know either, and I wouldn't be surprised if it's all wrong. The solution to the first one makes sense (when you know what it's supposed to do) but I still have a feeling that I'm just making humbug here.

Any help would be very much appreciated!