# Complex Analysis

Here, you find my whole video series about Complex Analysis in the correct order and I also help you with some text around the videos. If you want to test your knowledge, please use the quizzes, and consult the PDF version of the video if needed. When you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:

### Introduction

Complex analysis is a video series I started for everyone who is interested in calculus with the complex numbers and wants to expand her knowledge beyond the calculus with real numbers. Some basic facts from my Real Analysis course is needed but always mentioned in the videos.

With this you now know the foundations that we will need to start with this series. Some important bullet points are sequences, continuity, and derivatives for real functions. Then we will always expand the notions to the complex realm. Now, in the next video let us discuss what a derivative for a complex function is.

### Let’s get started

The notion of a derivative is fundamental in a lot of mathematical topics. In my real analysis course, you already learnt how to define it. It is literally the same for the complex functions. However, the conclusions from this definition we can form might be different. In the next video, we can immediately define what it means that a function is complex differentiable. Since this is a local property, we have to fix a point in the domain of the function. However, in order to be meaningful, the domain should be an open set. This is a general concept, we will also explain now:

Now we define the complex derivative for a function and explain the linear approximation we get from this. We also explain some examples:

Now we explain the terms holomorphic function and entire function:

Let us translate a complex function to a real function:

And now we are finally able to talk about the important Cauchy-Riemann equations:

Let’s now look at examples for the Cauchy-Riemann equations:

The following Wirtinger derivatives can be very helpful in quick calculations and give a nice short formula for the Cauchy-Riemann equations:

Now let us talk about power series:

When we talk about power series, the notion of uniform convergence is very important:

Let us prove this important fact about power series:

Now talk about Exp, Cos, and Sin as power series:

By knowing the exponential function, we can also define the inverse function, the logarithm. However, in the complex numbers, this is more complicated than in the real numbers.

After the definition of the logarithm, we are able to define the power in the complex numbers:

Let’s start with a very important concept: Laurent series. One could say that they generalise power series.

After knowing what Laurent series are, we are able to talk about another important concept: isolated singularities. These are also important for holomomorphic functions that are not defined on the whole complex plane.

Now we are finally ready to extend the integral into the complex realm. We start with defining the integral for functions on real intervals:

In the next step, we can talk about the important contour integral in the complex plane. We start with the definition and look at some simple examples.

Now we extend the complex integral to piecewise continuously differentiable paths and talk about important properties of the contour integral.

In the next video, we introduce primitives for complex-valued functions. A more suitable name is just antiderivatives. They can be used to calculate contour integrals.