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: $e^{iy} = \cos y + i \sin y$