This graph shows a set in the z-plane and its image under the exponential in a separate window. This is only one way to investigate a complex function with f(z).
It is easy to see that 0 is mapped onto 1 (on the real line). Note also how the image avoids the origin. (There is no log(0).) The graph also avoids "infinity" but this is not so obvious in this view. In fact, there is complete symmetry between 0 and infinity. (You can see this by asking yourself what is the effect of replacing z with -z in the domain. Recall that exp(-z) = 1/exp(z). )
A single mouse click redraws any figure on the Riemann sphere, where infinity is represented as a single point.
Be sure to check out the exponential in four dimensions.