ABSTRACT
One of the remarkable features of holomorphic function theory is that one can express the derivative of a holomorphic function in terms of the function itself. Nothing of the sort is true for real functions. One upshot is that one can obtain powerful estimates for the derivatives of holomorphic functions. This chapter discusses the Cauchy estimates and Liouville's theorem. One of the most elegant applications of Liouville’s Theorem is a verification of what is known as the Fundamental Theorem of Algebra. The chapter provides the sequences of holomorphic functions and their derivatives, the Power Series Representation of a Holomorphic Function, and a table of elementary power series. The zeros of a holomorphic function, uniqueness of analytic continuation, and the principle of persistence of functional relations are discussed.
