Book Recommendations for Picard Big and Little Theorems I am looking for a book covering Picard's Little and Big Theorems, preferably,  one that is intended for an undergraduate/first year graduate student who has a semester of complex analysis under his/her belt.
A big problem I have run into is that most books that I have found in my library prove only Picard's Little Theorem, so I especially need a readable proof of Picard's Big Theorem.
 A: Complex Made Simple by David Ullrich perfectly matches your request.  A good portion of the book seems to be written with the intent of leading a first year graduate student through Picard’s theorems.
Ullrich presents the proofs in all of their gory details including the subtle connections of Picard’s Theorems with hyperbolic geometry.  Other books intended for first year graduate students usually hide this from the reader.  Another big advantage of this book is that it simultaneously prepares the reader for both the Little Theorem and the Big Theorem.
Here are some of the topics involved/a general outline of the proofs employed by Ullrich:
In Complex Made Simple, Ullrich establishes that the Linear Fractional Transformations, with certain restrictions that you can read about, are the automorphisms of the upper-half plane (bijective holomorphic functions from the upper half plane to the upper half plane).  Ullrich also establishes that the automorphisms of the upper half plane are conjugate to the automorphisms of the unit disk (hence the connection with hyperbolic geometry).  Ullrich then uses a very carefully chosen domain and group to establish a function, $\lambda(s)$, that sends the upper-half plane to the $\mathbb{C}$\{0,1}. Ullrich then uses $\lambda(s)$ and a fact about holomorphic covering maps to prove Picard’s Little Theorem.
