Find-the-principal-value-of-3-4i-1-3-

Question Number 6923 by Tawakalitu. last updated on 03/Aug/16

F i n d t h e p r i n c i p a l v a l u e o f {(3 + 4 i)}^{\frac{1}{3}}

Commented by Yozzii last updated on 03/Aug/16

3 + 4 i = \sqrt{3^{2} + 4^{2}} e^{{i t a n}^{- 1} (4 / 3)} = 5 e^{{i t a n}^{- 1} \frac{4}{3}}

∴ p r i n c i p a l v a l u e o f {(3 + 4 i)}^{1 / 3} i s

{(5 e^{{i t a n}^{- 1} \frac{4}{3}})}^{1 / 3} = 5^{1 / 3} e^{i \frac{1}{3} {t a n}^{- 1} \frac{4}{3}} .

- - - - - - - - - - - - - - - - - - - - - -

z = {r e}^{i θ} = {r e}^{i (θ + 2 π k)} (k \in Z)

\Rightarrow z^{1 / n} = r^{1 / n} e^{i \frac{θ + 2 π k}{n}} (k = 0, 1, 2, \dots, n - 1)

p r i n c i p a l v a l u e o f z^{1 / n} o c c u r s a t k = 0 .

Commented by Tawakalitu. last updated on 03/Aug/16

I r e a l l y a p p r e c i a t e