Complexity

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))