Math2.org Math Tables: Complexity

(Math)

Basic Operations

\[i = \sqrt{-1}\]
\[i^2 = -1\]
\[1 / i = -i\]
\[i<sup>4k</sup> = 1; i<sup>4k+1</sup> = i; i<sup>4k+2</sup> = -1; i<sup>4k+3</sup> = -i
\, \text{(integer k)}\]
\[\sqrt{i} = \sqrt{1/2} + \sqrt{1/2} i\]

Complex Definitions of Functions and Operations

\[(a + bi) + (c + di) = (a+c) + (b + d) i\]
\[(a + bi) (c + di) = ac + adi + bci + bdi<sup>2</sup> = (ac - bd) + (ad +bc) i\]
\[\frac{1}{a + bi} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2} i\]
\[\frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i\]
\[a<sup>2</sup> + b<sup>2</sup> = (a + bi) (a - bi)\] (sum of squares)
\[e<sup>i \theta</sup> = \cos(\theta) + i \sin(\theta)\]
\[n<sup>a + bi</sup> = n^a (\cos(b \ln n) + i \sin(b \ln n)) \]
\[\text{If} \, z = r(\cos \theta + i \sin \theta) \,
\text{then} \, z<sup>n</sup> = r<sup>n</sup> ( \cos n \theta + i \sin n \theta ) \] (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
\[z(k) = r1/n [ \cos \frac{\theta + 2 \pi k}{n} + i \sin \frac{\theta + 2 \pi k}{n} ]\]
\[\text{If} \, z = r (\cos \theta + i \sin \theta) \,
\text{then} \, ln(z) = \ln r + i \theta\]
\[sin(a + bi) = sin(a)cosh(b) + cos(a)sinh(b) i\]
\[cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b) i\]
\[tan(a + bi) = \frac{\tan(a) + i \tanh(b)}{1 - i \tan(a) \tanh(b)} \] \[ = \frac{sech^2(b)\tan(a) + \sec^2(a)tanh(b) i}{1 + \tan^2(a)tanh^2(b)}\]