COMPLEX NUMBER
A. CARTESIAN AND POLAR FORM
| Cartesian Form | Polar Form | |
|---|---|---|
| Complex Number | \(z = x + y \: i\)
\(x = r \cos \theta\) \(y = r \sin \theta\) |
\(z = r \:.\: (\cos \theta + i \sin \theta) = r \:.\: cis \: \theta\)
\(\theta = \text{arg } z\) \(\tan \theta = \dfrac yx\), where \(-\pi < \theta < \pi\) |
| Conjugate | \(\overline z = x - y i\)
\(z \overline z = x^2 + y^2\) |
\(\overline z = r \:.\: (\cos \theta - i \sin \theta)\)
\(z \overline z = r^2\) |
| \(z^{-1} = \dfrac 1z\) | \(z_1 \:.\: z_2 = r_1 \:.\: r_2 \:.\: cis \: (\theta_1 + \theta_2)\)
\(\dfrac {z_1}{z_2} = \dfrac{r_1}{r_2} \:.\: cis \: (\theta_1 - \theta_2)\) |
B. EXPONENTIAL FORM
\(z = r \: e^{i \: \theta}\)
\(z_1 \:.\: z_2 = r_1 \:.\: r_2 \: e^{i \: (\theta_1 + \theta_2)}\)
C. DE MOIVRE'S THEOREM
\(z = r \:.\: (\cos \theta + i \sin \theta)\)
\(z^n= r^n \:.\: (\cos n \theta + i \sin n \theta)\)
\(z^{-n}= r^{-n} \:.\: (\cos n \theta - i \sin n \theta)\)
\(z^{\frac 1n}= r^{\frac 1n} \:.\: \left[\cos \left(\dfrac {\theta + 2 \pi k}{n} \right) + i \sin \left(\dfrac {\theta + 2 \pi k}{n} \right) \right]\)
D. GEOMETRY OF COMPLEX NUMBER