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Complex Analysis

Table of Contents

1. Idea

We use complex variables \(z=x+\I y\), then do calculus with it, and we transform 2-dimensional problems into something as simple as one-dimensional calculus.

2. Derivatives

The differential operators we work with are written in the coordinates \(z=x + \I y\) and its complex conjugate \(\bar{z} = x - \I y\). The derivatives in these coordinates may be written as

\begin{equation} \frac{\partial}{\partial z} = \frac{1}{2} \left(\frac{\partial}{\partial x} - \I\frac{\partial}{\partial y}\right) \end{equation}

and

\begin{equation} \frac{\partial}{\partial \bar{z}} = \frac{1}{2} \left(\frac{\partial}{\partial x} + \I\frac{\partial}{\partial y}\right). \end{equation}

Often these derivatives are written \(\partial\) and \(\bar{\partial}\).

I've seen these called Wirtinger derivatives, but I thought Wirtinger derivatives were operators on functions of specifically multiple complex variables.

We call a complex function \(f(z,\bar{z})\) Holomorphic if

\begin{equation} \frac{\partial}{\partial\bar{z}}f(z,\bar{z}) = 0 \end{equation}

everywhere.

Last Updated 2021-06-01 Tue 10:00.