Basic Operations
\[i = \sqrt{-1}\]
\[i^2 = -1\]
\[1 / i = -i\]
\[i^{4k} = 1; i^{4k+1} = i; i^{4k+2} = -1; i^{4k+3} = -i
\, \text{(integer k)}\]
\[\sqrt{i} = \sqrt{1/2} + \sqrt{1/2} i\]
Complex Definitions of Functions and Operations
\[(a + bi) + (c + di) = (a+c) + (b + d) i\]
\[(a + bi) (c + di) = ac + adi + bci + bdi^{2} = (ac - bd) + (ad +bc) i\]
\[\frac{1}{a + bi} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2} i\]
\[\frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i\]
\[a^{2} + b^{2} = (a + bi) (a - bi)\] (sum of squares)
\[e^{i \theta} = \cos(\theta) + i \sin(\theta)\]
\[n^{a + bi} = n^a (\cos(b \ln n) + i \sin(b \ln n)) \]
\[\text{If} \, z = r(\cos \theta + i \sin \theta) \,
\text{then} \, z^{n} = r^{n} ( \cos n \theta + i \sin n \theta ) \]
(DeMoivre's Theorem)
if \[w = r(\cos \theta + i \sin \theta);n=integer\]
then there are n complex nth roots (z) of w for k=0,1,..n-1
\[z(k) = r^{1/n} [ \cos \frac{\theta + 2 \pi k}{n} + i \sin \frac{\theta + 2 \pi k}{n} ]\]
\[\text{If} \, z = r (\cos \theta + i \sin \theta) \,
\text{then} \, ln(z) = \ln r + i \theta\]
\[sin(a + bi) = sin(a)cosh(b) + cos(a)sinh(b) i\]
\[cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b) i\]
\[tan(a + bi) = \frac{\tan(a) + i \tanh(b)}{1 - i \tan(a) \tanh(b)} \]
\[ = \frac{sech^2(b)\tan(a) + \sec^2(a)tanh(b) i}{1 + \tan^2(a)tanh^2(b)}\]
