💡 We define a new number, namely $i$, that solves the equation $x^2+1=0$. We will call this number the imaginary unit and assign the value $i=\sqrt{-1}$.

💡 A complex number $z$ is a number of the form $z=x+iy$, where $x$ and $y$ are real numbers and $i$ the imaginary unit. $x$ is called the real part and is denoted $x=\text{Re}(z)$ $y$ is called the imaginary part and is denoted $y=\text{Im}(z)$

💡 The set of all complex numbers is called $\mathbb{C}=\{z=x+iy|x,y\in \mathbb{R}\}$.

💡 Numbers of the form $iy$ are called purely imaginary.

💡 Addition of two complex numbers $z=x+iy$ and $w=u+iv$:

$$z+w:=(x+u)+i(y+v).$$

💡 Multiplication of a complex number $z=x+iy$ with a real number $\alpha$:

$$\alpha z:=\alpha x+i\alpha y.$$

💡 Multiplication of two complex numbers $z=x+iy$ and $w=u+iv$:

$$wz:=(u+iv)(x+iy)=ux+iuy+ivx+i^{2}vy=(ux-vy)+i(uy+vx).$$

💡 Let $z=x+iy$ be a complex number. The conjugate to $z$ is $\bar{z}:=x-iy$. Then:

$$z\bar{z}=x^2+y^2\ge 0$$

💡 The non-negative real number $|z|:=\sqrt{z\bar{z}}$ is called the absolute value or the modulus of the complex number $z$.

💡 Division of two complex number $z$ and $w$:

$$\frac{z}{w}:=\frac{z\bar{w}}{w\bar{w}}=\frac{z\bar{w}}{|w{|}^2}$$

💡 Some useful rules

• $\overline{zw}=\bar{z}\cdot \bar{w}$
• $\overline{(\frac{z}{w})}=\frac{\bar{z}}{\bar{w}}$
• $\overline{\stackrel{ˉ}{z}}=z$
• $|zw|=|z||w|$
• $|\frac{z}{w}|=\frac{|z|}{|w|}$

💡 Triangle Inequality:

$$|z+w|\le |z|+|w|$$

💡 Polar Form:

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where $r=|z|=\sqrt{x^2+y^2}$ is the distance to the origin and $\theta$ is the angle between the $x$-axis and the position vector to $z$. The latter is also described as the argument, angle or phase of $z$.

Since $\theta$ has a period of $2\pi$, i.e., $\theta =\theta +2\pi =\theta -2\pi =\dots$ the polar form representation of a complex number is not unique. For this reason $\theta$ is bound to the interval $[0,2\pi )$, the so-called principal value of the argument.

💡 Trigonometric Form:

$$z=x+iy=r(\cos \theta +i\sin \theta )=r\cdot \text{cis}\theta$$

💡 Trigonometry Recap:

• $\cos \theta =\frac{x}{|z|}$
• $\sin \theta =\frac{y}{|z|}$
• $\tan \theta =\frac{y}{x}$

💡 Euler’s Formula:

$$e^{i\theta }=\cos \theta +i\sin \theta$$

This formula let’s us write a complex number $z$ in exponential form:

$$z=x+iy=r\cdot \text{cis}\theta =r{e}^{i\theta }$$

For $z=0$, it holds that $r=|z|=0$ and the argument can be chosen freely.

💡 Useful rules for exponential form:

• $e^{i\theta }\cdot e^{i\alpha }=e^{i(\theta +\alpha )}$
• $\frac{e^{i\theta }}{e^{i\alpha }}=e^{i(\theta -\alpha )}$
• Let $z=r\cdot e^{i\theta }$ and $w=s\cdot e^{i\alpha }$. Then $zw=rs{e}^{i(\theta +\alpha )}$ and $\frac{z}{w}=\frac{r}{s}{e}^{i(\theta -\alpha )}$, $w\ne 0$
• $e^{2\pi i}=\cos 2\pi +i\sin 2\pi =1$
• Let $z=r\cdot e^{i\theta }$. Then $z^n=r^n{e}^{i\theta n}$

💡 Euler’s Identity:

$$\begin{array}{rl}& e^{i\pi }=\cos \pi +i\sin \pi =-1\\ ⟹& e^{\pi i}+1=0\end{array}$$

💡 Switching from polar to normal form: Let $z=r\cdot e^{i\theta }=x+iy$. Then $x=r\cdot \cos \theta$ and $y=r\cdot \sin \theta$.

💡 Switching from normal to polar form: Let $z=x+iy=r\cdot e^{i\theta }$. Then $r=|z|=\sqrt{x^2+y^2}$. To determine the argument $\theta$ we need to differentiate the following cases: