I am in need of a specific (simple, if possible) complex bijection mapping that would map a square onto the unit disk, including an explanation/examination of "why it works". I need it as an example showing the Riemann theorem in practice, yet despite thinking/trying and searching online, I cannot come up with a good idea. If possible, additional explanation of how the mapping is constructed/which way of thinking about it yields the result would be very much appreciated, so that I can understand how these problems can be approached on a general basis. Thank you kindly.
edit: I've found this https://mathematica.stackexchange.com/questions/159067/schwarz-christoffel-maps-from-unit-disk-to-regular-polygons-visualization; if I plug in n = 4 here, would I get what I'm looking for, and if yes, how do I prove "it works", that is, that it is a bijection, conformal, and that it indeed maps the unit disk onto a square?
Also a note: I am a beginner with Complex Analysis, so apologies if these questions are trivial.