The main goal of this topic is to define and give some of the important properties of complex analytic functions. A function \(f(z)\) is analytic if it has a complex derivative \(f'(z)\). In general, the rules for computing derivatives will be familiar to you from single variable calculus. However, a much richer set of conclusions can be drawn about a complex analytic function than is generally true about real differentiable functions.
- 2.1: The Derivative - Preliminaries
- In calculus we defined the derivative as a limit. In complex analysis we will do the same. Before giving the derivative our full attention we are going to have to spend some time exploring and understanding limits.
- 2.6: Cauchy-Riemann Equations
- The Cauchy-Riemann equations are our first consequence of the fact that the limit defining f(z) must be the same no matter which direction you approach z from. The Cauchy-Riemann equations will be one of the most important tools in our toolbox.
- 2.8: Gallery of Functions
- In this section we’ll look at many of the functions you know and love as functions of z . For each one we’ll have to do four things. (1) Define how to compute it. (2) Specify a branch (if necessary) giving its range. (3) Specify a domain (with branch cut if necessary) where it is analytic. (4) Compute its derivative.
- 2.9: Branch Cuts and Function Composition
- We often compose functions, i.e. f(g(z)) . In general in this case we have the chain rule to compute the derivative. However we need to specify the domain for z where the function is analytic. And when branches and branch cuts are involved we need to take care.