According to the Inverse Function Theorem, for every [tex]z_0 \in C[/tex] there exists r > 0 such that the exponential function [tex]f(z) = e^z[/tex] maps D(z0; r) invertibly to an open set [tex]U = f(D(z_0; r))[/tex]. (a) Find the largest value of r for which this statement holds, and (b) determine the corresponding open set U in case z0 = 0: For part (b) use Mathematica's ParametricPlot or ContourPlot functions..
The Attempt at a Solution
I can't find anything in my book that remotely tells me how to go about finding r. I've been rather bad at this class and I would appreciate if someone could point me out to better books on complex analysis that are not Marsden's Basic Complex Analysis. I feel like I get very hard questions and they are usually like this one and I find the book to be quite hard to decipher and understand.
I would like to know first how this mapping is determined. The function e^z takes a disk of some radius centered at z0 and maps it into an open set. I don't know what to make of it, is that a very generic description or something? In any case, what kind of limitations am I supposed to be thinking, what kind of conditions should there be to figure out what the largest r would be and ultimately, how do you find this r?