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 Complexity (Math | Miscellaneous | Complexity)

### Basic Operations

i = (-1)

i 2 = -1

1 / i = -i

i 4k = 1; i (4k+1) = i; i (4k+2) = -1; i (4k+3) = -i (k = integer)

( i ) = (1/2)+ (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

1/(a + BI) = a/(a 2 + b 2) - b/(a 2 + b 2) i

(a + BI) / (c + DI) = (ac + BD)/(c 2 + d 2) + (BC - ad)/(c 2 +d 2) i

a2 + b2 = (a + BI) (a - BI)   (sum of squares)

e (i ) = cos + i sin

n (a + BI) = (COs(b ln n) + i sin(b ln n))n a

if z = r(COs + i sin ) then z n = r n ( COs n+ i sin n )(DeMoivre's Theorem)

if w = r(COs + i sin );n=integer, then there are n complex nth roots (z) of w for k=0,1,..n-1:

z(k) = r (1/n) [ COs( (+ 2(PI)k)/n ) + i sin( (+ 2(PI)k)/n ) ]

if z = r (COs + i sin ) then ln(z) = ln r + i

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) = ( tan(a) + i tanh(b) ) / ( 1 - i tan(a) tanh(b))
= ( sech 2(b)tan(a) + sec 2(a)tanh(b) i ) / (1 + tan 2(a)tanh 2(b))