Add overview to complex

src/1_complex_numbers.md

@@ -4,7 +4,11 @@ title: Complex Numbers
 
 # Complex numbers
 
-The lecture on complex numbers consists of three parts, each with their own video and text.
+The lecture on complex numbers consists of three parts, each with their own video:
+
+- [Definition and basic operations](#definition-and-basic-operations)
+- [Complex functions](#complex-functions)
+- [Differentiation and integration](#differentiation-and-integration)
 
 **Total video length: 38 minutes and 53 seconds**
 
@@ -189,7 +193,7 @@ $$\begin{align} z = e^{{\rm i} \varphi} & \Rightarrow z^4 = e^{4{\rm i} \varphi}
 & \Leftrightarrow \varphi = 0, \varphi = \frac{\pi}{2}, \varphi = -\frac{\pi}{2}, \varphi = \pi \\
 & \Leftrightarrow z = 1, z = i, z = -i, z = -1 \end{align}$$
 
-### Differentiation and integration
+## Differentiation and integration
 
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/JyftSqmmVdU" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 
@@ -198,6 +202,8 @@ We only consider differentiation and integration over *real* variables. We can t
 $$\frac{d}{d\varphi} e^{{\rm i} \varphi} = e^{{\rm i} \varphi} \frac{d}{d\varphi} ({\rm i} \varphi) ={\rm i} e^{{\rm i} \varphi} .$$
 $$\int_{0}^{\pi} e^{{\rm i} \varphi} = \frac{1}{{\rm i}} \left[ e^{{\rm i} \varphi} \right]_{0}^{\pi} = -{\rm i}(-1 -1) = 2 {\rm i}$$
 
+## Bonus: the complex exponential function and trigonometry
+
 Let us show some tricks where the simple properties of the exponential
 function helps in re-deriving trigonometric identities.