(0:00) Exam information. (1:28) "Holomorphic function" means "analytic function"...we'll look at "meromorphic functions" as well. (2:26) A function that is analytic at a point has a Taylor series that converges to it in the largest open disk centered at the point over which the function is analytic. (5:32) Given a power series with a positive radius of convergence, it defines an analytic function whose Taylor series is the given power series. (7:30) Some people define analyticity in terms of power series. This also can be done to define real analytic functions, though that is not equivalent to being infinitely differentiable. (9:04) Laurent series representation in an annulus.
Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.