06 Intro to Complex Calculus
Complex Numbers¶
\[ z = x + iy \]
Make sure that all calculations are in radian
Properties¶
\[ \begin{aligned} |z| &= \sqrt{x^2 + y^2} \\ |z_1 \cdot z_2| &= |z_1| \cdot |z_2| \\ \left| \frac{z_1}{z_2} \right| &= \frac{ |z_1| }{ |z_2| } \\ \bar z &= x - iy \\ |\bar z| &= |z| \\ \bar{ |z| }^2 &= z \cdot \bar z \\ \frac{z + \bar z}{2} &= \text{Re}(z) \\ \frac{z - \bar z}{2i} &= \text{Im}(z) \\ \overline{z_1 \pm z_2} &= \bar z_1 \pm \bar z_2 \\ \overline{z_1 \cdot z_2} &= \bar z_1 \cdot \bar z_2 \\ \overline{\left( \frac{z_1}{z_2} \right)} &= \frac{\bar z_1}{\bar z_2} \end{aligned} \]
Circles¶
| \(\vert z \vert = r\) | circle with radius \(r\) @ \((0, 0)\) |
| \(\vert z-z_0 \vert = r\) | circle with radius \(r\) @ \(z_0\) |
Triangle Inequality¶
| Upper Bound | Lower Bound |
|---|---|
| \(\vert z_1 \pm z_2 \vert \le \vert z_1 \vert + \vert z_2 \vert\) | \(\vert z_1 \pm z_2 \vert \ge \text{abs} (\vert z_1 \vert - \vert z_2 \vert )\) |
abs refers to absolute value
Argument¶
\[ \begin{aligned} \text{arg } z &= \left| \frac{y}{x} \right| \\ \text{Arg } z &= \text{Principle Value of arg } z\\ \text{arg}(z_1 \cdot z_2) &= \text{arg}(z_1) + \text{arg}(z_2) \\ \text{arg}\left( \frac{z_1}{z_2} \right) &= {\text{arg}(z_1)} - {\text{arg}(z_2)} \end{aligned} \]
Polar Form¶
\[ \begin{aligned} z &= r \cdot e^{i \theta} \\ &= r (\cos \theta + i \sin \theta) \end{aligned} \]
Root¶
\[ \begin{aligned} c &= (r \cdot e^{i\theta})^{\frac{1}{n}} \\ &= r^{\frac{1}{n}} \cdot e^{\frac{i\theta}{n}} \\ &= r^{\frac{1}{n}} \Bigg( \cos \left(\frac{\theta}{n}\right) + i \sin \left(\frac{\theta}{n}\right) \Bigg) \\ r &= |z| \\ \frac{\theta}{n} &= \frac{\text{Arg }z + 2k\pi}{n}, k \in [0, n) \\ e^{i(n\theta)} &= \cos(n\theta) + i \sin(n\theta) \\ e^{-i(n\theta)} &= \cos(n\theta) - i \sin(n\theta) \end{aligned} \]