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 ) thenln(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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