Números complejos

Operaciones básicas

i^{^2} = -1

1 / i = -i

i^{^(4k)} = 1; i^{^(4k+1)} = i; i^{^(4k+2)} = -1; i^{^(4k+3)} = -i (k = un entero)

( i ) = (1/2)+ (1/2) i

Definiciones de funciones y operaciones complejas

(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)   (la suma de cuadrados)

e^{^(i )} = cos + i sen

n^{^(a + bi)} = (cos(b ln n) + i sen(b ln n))n^{^a}

si z = r(cos + i sen ) pues z^{^n} = r^{^n} ( cos n+ i sen n )(El Teorema de DeMoivre)

si w = r(cos + i sen ); n=un entero. entonces hay n, n^th raices complejas (z) de w para k=0,1,..n-1:

z(k) = r^{^(1/n)} [ cos( (+ 2(PI)k)/n ) + i sen( (+ 2(PI)k)/n ) ]

si z = r (cos + i sen ) pues ln(z) = ln r + i

sen(a + bi) = sen(a)cosh(b) + cos(a)senh(b) i

cos(a + bi) = cos(a)cosh(b) - sen(a)senh(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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