Considering $ (1+i)^n - (1 - i)^n $, Complex Analysis I have been working on problems from Complex Analysis by Ahlfors, and I got stuck in the following problem:
Evaluate:
$$
(1 + i)^n - (1-i)^n 
$$
I have just "reduced" to:
$$ 
(1 + i)^n - (1-i)^n  = \sum_{k=0} ^n i^k(1 - (-1)^k)
$$
by using expansion of each term.
Thanks.

A complex number has both a modulus and angle. The modulus is the norm. Let $z=x+yi$. Then the modulus is
$$
\lvert z\rvert =\sqrt{x^2+y^2}
$$
and the angle is 
$$
\phi = \arctan(y/x).
$$
Now you can write the Complex number in exponential form
$$
z=\lvert z\rvert e^{i\phi}.
$$
You should be able to take it from here once you convert the complex numbers to exponential form. Remember to pay attention to the quadrant when you take the arctan.
