Basic Operations

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))

 
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