I was asked if it is possible to solve this question with the complex exponential:
$\displaystyle (z + i)^5 - (z-i)^5=0$
I can see how this is solved by expanding out, but is there an easier or more systematic way to go about these problems?
I was asked if it is possible to solve this question with the complex exponential:
$\displaystyle (z + i)^5 - (z-i)^5=0$
I can see how this is solved by expanding out, but is there an easier or more systematic way to go about these problems?
Well it is
$\displaystyle (z+i)^5=(z-i)^5$
Taking fifth roots on both sides, we have
$\displaystyle \omega(z+i)=z-i$
where $\displaystyle \omega^5=1$ is any fifth root of unity. Thus
$\displaystyle z(1-\omega)=i(1+\omega)$
or $\displaystyle z=\frac{i(1+\omega)}{1-\omega}$.