# 2: Analytic Functions

- Page ID
- 6479

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

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.