9.6: Further remarks
In Chapter 5 we defined cosine and sine in terms of power series. In Section 9.2, we interpreted them geometrically and used trigonometric identities. Showing that the power series and the trigonometric interpretation are really describing the same function is part of a course in Complex Analysis.
There are two main ingredients to a first course in Complex Analysis. The first is to show that if a function \(f\) has a derivative everywhere on some open disk, in the sense that \[\lim _{z \rightarrow z_{0}} \frac{f\left(z_{0}\right)-f(z)}{z_{0}-z}\] exists, then the function is automatically analytic, i.e. expressible by a convergent power series. This is not true for real functions, and explains much of the special nature of complex differentiable functions.
The second part of the course concerns evaluating contour integrals of complex differentiable functions. This is useful not only in its own right, but in applications to real analysis, such as inverting the Laplace transform, or evaluating definite integrals.
A good introduction to Complex analysis is the book by Donald Sarason \([7]\) .