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Chapter 2: Q16P (page 74)

Evaluate each of the following in x+ iyform, and compare with a computer solution.
${(\frac{1 + i \sqrt{3}}{2})}^{'}$

Short Answer

Expert verified

The value of $(\frac{1 + i \sqrt{3}}{2})$ is Where $n = 0, 1, 2, 3 . .$ .

Step by step solution

01

Given Information.

The given expression is ${(\frac{1 + i \sqrt{3}}{2})}^{'}$ .

02

Definition of Complex Number.

The numbers that are presented in the form of $x + i y$ where, $' x, y'$ are real numbers and $' i'$ is an imaginary number, those numbers are referred to as called Complex numbers.

03

Find the value of (1+i32)'.

Let the complex numbers as:

$u = {(\frac{1 + i \sqrt{3}}{2})}^{'} z = \frac{1 + i \sqrt{3}}{2}$

Find the modulus of the complex number.

$r = |z| = 1$ .

Find the argument of the complex number.

$胃 = \frac{蟺}{3} u = e^{(In {(z)}^{i})} e^{(I n (z) i)} = e^{(i [In (r) + i (胃 + 2 n 蟺)])} = e^{[i (I n (1) + i (\frac{蟺}{3} + 2 n 蟺))]} = e^{[- (\frac{蟺}{3} + 2 n 蟺)]} n = 0, 1, 2, 3, . . .$

Hence the value of ${(\frac{1 + i \sqrt{3}}{2})}^{'}$ is $e^{[(- \frac{蟺}{3} + 2 苍蟺)]}$ Where role="math" localid="1658815979004" $n = 0, 1, 2, 3, . .$

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Most popular questions from this chapter

Describe geometrically the set of points in the complex plane satisfying the following equations. $| z | = 2$ .

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

$\frac{2 i - 1}{i - 2}$ .

Solve for all possible values of the real numbersand in the following equations ${(x + i y)}^{2} = {(x - i y)}^{2}$ .

Describe geometrically the set of points in the complex plane satisfying the following equations.

$R e (z^{2}) = 4$ .

Solve for all possible values of the real numbers $x$ and $y$ in the following equations. $(2 x - 3 y - 5) + i (x + 2 y + 1) = 0$

See all solutions

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