Answer to Question #116825 in Complex Analysis for desmond

Question

Express −1 + i in polar form. Hence show that (−1 + i)16 is real and that 1/(−1 + i)6
is purely imaginary, giving the value for each.

Answer 1

$z=-1+i=|z|(cos\theta+isin\theta)$

$|z|=\sqrt{1+1}=\sqrt{2}$

$cos\theta=-1/\sqrt{2}, sin\theta=1/\sqrt{2}, \theta=3\pi/4$

$-1+i=\sqrt{2}(cos\frac {3\pi}{4}+isin\frac {3\pi}{4})$

$z^n=|z|^n(cos(n\theta)+isin(n\theta))$

$(-1+i)^{16}=(\sqrt{2})^{16}(cos(\frac {16\cdot3\pi}{4})+isin(\frac {16\cdot3\pi}{4})=$

$=2^8(cos12\pi+isin12\pi)=256(1+0)=256$

$\frac {1}{(-1+i)^6}=\frac {1}{(\sqrt{2})^6(cos(\frac {6\cdot3\pi}{4})+isin(\frac {6\cdot3\pi}{4})}=$

$=\frac {1}{2^3(cos(9\pi/2)+isin(9\pi/2))}=\frac {1}{8(0+i)}=\frac {1}{8i}=-i/8$