# Thread: Domain and Range of Complex Functions

1. ## Domain and Range of Complex Functions

Determine the domain and range of the complex function:
$\displaystyle f(z)=e^{(x-y)\mathbf{i}}$

I know that forum wish that questions have worked associated with what one needs help with; however, my book doesn't help whatsoever in finding the domain and range. Therefore, I can't supply any work since I have no idea how do this problem. Can someone help me with solving this problem, thanks.

2. The function

$\displaystyle e^{i\theta}$ for real theta is always on the unit circle. So in your case you have $\displaystyle \theta=x-y$ So x and y can take on any values but the the immage will always lie on the unit circle in $\displaystyle \mathbb{C}$

3. The domain then is $\displaystyle \mathbb{C}$ and the range is simple the unit circle.

4. How can I determine a number w such that w isn't in the range?