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:

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.

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: