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.


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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:


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


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Now we explain the terms holomorphic function and entire function:


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Let us translate a complex function to a real function:


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


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


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


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Now let us talk about power series:


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


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Let us prove this important fact about power series:


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


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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.


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


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


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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.


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


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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.


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


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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.


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Building on the former video, we can also look at the converse statement. Can we conclude from the fact that contour integrals along closed curves are zero that an antiderivative has to exist?


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Now, we are ready to formulate and prove the famous integral theorem known as Goursat’s Theorem.


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So let’s generalise the last result and formulate Cauchy’s theorem for discs.


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In the next video, let’s go back to curves in the complex plane. We have learnt with Cauchy’s theorem that it is important to know if a curve lies in a disc for the domain of definition of a function. This guarantees that no point outside of the domain gets surrounded by the curve. This is essential for applying Cauchy’s theorem like the counterexample of the function $ f(z) = \frac{1}{z} $ shows us. Therefore, in order to generalise the theorem, we need to talk about winding number for curves and points.


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After learning how Cauchy’s integration theorem works for discs, we can generalise it two more general domains. In particular, we can explicitly prove Cauchy’s Theorem for more domains:


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In the following video, we will talk about a very special contour integral. We call it a keyhole contour because it looks like it. By using Cauchy’s integral theorem, we can show a very important fact for holomorphic functions with an isolated singularity.


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And now finally, we can prove one of the most important formulas in Complex Analysis: Cauchy’s Integral Formula


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In the next part, we will generalise Cauchy’s integral formula also for derivatives. While doing this, we also show that each holomorphic can locally be represented by power series. In particular, this shows that each holomorphic function is a $ C^\infty $-function.


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The following result is very famous. It’s Liouville’s Theorem. It states that each entire is either constant or unbounded. In order to prove this fact, we will need Cauchy’s inequalities, which explain how much the derivatives of holomorphic functions can grow.


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