Complex Variables and Partial Differential Equations (3130005)

Question-2c( i )-OR

BE | Semester-3 Summer-2020 | 27-10-2020

Q2) (c( i ))

Find the fourth roots of -1.

Let,

z = - 1

\Rightarrow x = - 1 & y = 0

\Rightarrow r = 1 & θ = π + 2 kπ

The polar form of complex number,

- 1 = r (cosθ + i sinθ)

\Rightarrow - 1 = 1 (\cos (π + 2 kπ) + i \sin (π + 2 kπ))

\Rightarrow {(- 1)}^{\frac{1}{4}} = {[(\cos (π + 2 kπ) + i \sin (π + 2 kπ))]}^{\frac{1}{4}}

\Rightarrow {(- 1)}^{\frac{1}{4}} = [(\cos (\frac{π + 2 kπ}{4}) + i \sin (\frac{π + 2 kπ}{4}))] (De Movire' s Theorem)

; k = 0, 1, 2, 3 .

If

k = 0

, then

z_{1} = \cos \frac{π}{4} + i \sin \frac{π}{4}

.

If

k = 1

, then

z_{2} = \cos \frac{3 π}{4} + i \sin \frac{3 π}{4}

.

If

k = 2

, then

z_{3} = \cos \frac{5 π}{4} + i \sin \frac{5 π}{4}

.

If

k = 3

, then

z_{4} = \cos \frac{7 π}{4} + i \sin \frac{7 π}{4}

.

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