diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md index ae0fa27bfbbcde8a20f1f692c5ae0b56fa890e3c..95f11c4ad15debcb61668d3f6d19ae65914a24eb 100644 --- a/src/1_complex_numbers.md +++ b/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.