OPERACIONES CON COMPLEJOS





1) SUMA

z_{1}=a+bi
z_{2}=c+di

z_{1}+z_{2}=(a+c)+(b+d)i

2) MULTIPLICACIÓN

    (a+bi)(c+di)=ac+adi+bci+bdi^{2}

\therefore    (a+bi)(c+di)=(ac-bd)+(ad+bc)i

En forma polar:

z_{1}=a+bi=\rho_{1}(\cos\theta_{1}+i\sin\theta_{1})
z_{2}=c+di=\rho_{2}(\cos\theta_{2}+i\sin\theta_{2})

z_{1}.z_{2}=\rho_{1}(\cos\theta_{1}+i\sin\theta_{1})\rho_{2}(\cos\theta_{2}+i\sin\theta_{2})

=\rho_{1}.\rho_{2}\left[(\cos\theta_{1}\cos\theta_{2}-\sin\theta_{1}\sin\theta_{2})+i(\sin\theta_{1}\cos\theta_{2}+\cos\theta_{1}\sin\theta_{2})\right]

z_{1}.z_{2}=\rho_{1}.\rho_{2}[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})]

3) DIVISIÓN

Dividir z_{1}:z_{2}, sí:

z_{1}=a+bi
z_{2}=c+di

\dfrac{z_{1}}{z_{2}}=\dfrac{a+bi}{c+di}=\dfrac{(a+bi)(c-di)}{(c+di)(c-di)}=\dfrac{(ac+bd)+(bc-ad)i}{c^{2}-d^{2}i^{2}}

=\dfrac{(ac+bd)}{c^{2}+d^{2}}+\dfrac{(bc-ad)i}{c^{2}+d^{2}}

Forma polar:

\dfrac{z_{1}}{z_{2}}=\dfrac{\rho_{1}(\cos\theta_{1}+i\sin\theta_{1})}{\rho_{2}(\cos\theta_{2}+i\sin\theta_{2})}.\dfrac{(\cos\theta_{2}-i\sin\theta_{2})}{(\cos\theta_{2}-i\sin\theta_{2})}

\dfrac{z_{1}}{z_{2}}=\dfrac{\rho_{1}}{\rho_{2}}\left[\cos\left(\theta_{1}-\theta_{2}\right)+i\sin\left(\theta_{1}-\theta_{2}\right)\right]

4) POTENCIA.- Fórmula de Moivre:

\left[\rho(\cos\theta+i\sin\theta)\right]^{n}=\rho^{n}\left(\cos n\theta+i\sin n\theta\right)

5) RAÍZ

\sqrt[n]{\rho(\cos\theta+i\sin\theta)}=\sqrt[n]{\rho}\left[\cos\left(\dfrac{\theta+2k\pi}{n}\right)+i\sin\left(\dfrac{\theta+2k\pi}{n}\right)\right]

Donde K=0,1,2,3,\ldots,(n-1)



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