Hi guys, need help with this question:
Show that the roots of the EQN z^5 - (z - i)^5 = 0, z is not equal to i, are
(1/2)[cot(k(pi)/5) + i] where k = 1,2,3,4.
Thanks in advance!
Try this line of reasoning:
$\displaystyle z^{5}-(z-i)^{5}=0$
$\displaystyle z^{5}=(z-i)^{5}$
$\displaystyle \left(\frac{z-i}{z}\right)^{\!\!5}=1.$
Let
$\displaystyle w=\frac{z-i}{z}.$
Then $\displaystyle w^{5}=1,$ the fifth roots of unity (exactly). So I would try writing down the $\displaystyle w$'s, and then solving $\displaystyle zw=z-i$ for $\displaystyle z$ once you've done that.
Make sense?