Complex numbers

• $i^2 = -1$
• Complex number: $z = a + bi$
• Complex conjugate: $\overline{z} = a - bi$
• Modulus/absulute value: $|z|=\sqrt{a^2 + b^2}$
• Forms
  • Basic: $z = a + bi$
  • Polar: $z = r (\cos \theta + i \sin \theta)$
    • $r = |z| = \sqrt{a^2 + b^2}$
    • $\theta = \arg(z) = \arctan\left(\frac{b}{a}\right)$ (only on $[0,\frac{\pi}{2}]$)
  • Exponential: $z = re^{i\theta}$
• De Moivre's Theorem (powers): $z^n = r^n (\cos n \theta + i \sin n \theta)$
• Complex roots: $w_k = r^{1/n}(\cos(\frac{\theta + k\cdot2\pi}{n})+ i\sin(\frac{\theta + k\cdot2\pi}{n}))$
  • Has $n$ solutions
  • All $nth$ roots lie on a circle with radius $|w_k|=r^{1/n}$ (equally spread out).
• Euler's formula: $e^{iy} = \cos y + i \sin y$