Question

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Let's say a complex number is defined as, $z = i^i$.
Then find the value of, magnitude of the complex number $|z|$ and the amplitude of the complex $Amp(z)$.
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Answer 1

Answer:

Amp(z) = -π/2

|z| = -1

Step-by-step explanation:

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Answer 2

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:z = {i}^{i}$

So, let we express i in its polar form.

Let assume that

$\rm :\longmapsto\:i = r(cosx + i \: sinx) - - - (1)$

$\rm :\longmapsto\:i = rcosx + i \:r sinx$

So on comparing the real and Imaginary parts, we get

$\rm :\longmapsto\:rcosx = 0 - - - - (2)$

and

$\rm :\longmapsto\:r \: sinx = 1 - - - (3)$

On squaring equation (2) and (3) and adding, we get

$\rm :\longmapsto\: {r}^{2} {cos}^{2}x + {r}^{2} {sin}^{2}x = 1$

$\rm :\longmapsto\: {r}^{2} ({cos}^{2}x + {sin}^{2}x) = 1$

$\rm :\longmapsto\: {r}^{2} = 1$

$\bf\implies \:\boxed{ \tt{ \: r = 1 \: }}$

On substituting r = 1 in equation (2) and (3), we get

$\rm :\longmapsto\:cosx = 0$

and

$\rm :\longmapsto\:sinx = 1$

$\bf\implies \:\boxed{ \tt{ \: x \: = \: \dfrac{\pi}{2} \: \: }}$

On substituting the values of r and x, in equation (1), we get

$\rm :\longmapsto\:i = 1\bigg(cos\dfrac{\pi}{2} + isin\dfrac{\pi}{2} \bigg)$

$\bf\implies \:i = cos\dfrac{\pi}{2} + \: i \: sin\dfrac{\pi}{2}$

Hence, in exponential form, it is represented as

$\bf\implies \:i \: = \: {\bigg(e \bigg) }^{i\dfrac{\pi}{2} }$

Hence,

$\bf\implies \: {i}^{i} \: = \: {\bigg(e \bigg) }^{i\dfrac{\pi}{2} \times i}$

$\bf\implies \: {i}^{i} \: = \: {\bigg(e \bigg) }^{ {i}^{2} \dfrac{\pi}{2}}$

$\bf\implies \: {i}^{i} \: = \: {\bigg(e \bigg) }^{ \: - \: \dfrac{\pi}{2}}$

can be further rewritten as

$\bf\implies \: {i}^{i} \: = \: {\bigg(e \bigg) }^{ \: - \: \dfrac{\pi}{2}} \times 1$

$\bf\implies \: {i}^{i} \: = \: {\bigg(e \bigg) }^{ \: - \: \dfrac{\pi}{2}} \times (1 + 0i)$

$\bf\implies \: {i}^{i} \: = \: {\bigg(e \bigg) }^{ \: - \: \dfrac{\pi}{2}}\bigg(cos0 + i \: sin0\bigg)$

Hence,

$\rm \implies\:\boxed{ \tt{ \: arg(z) = 0 \: }}$

and

$\rm \implies\:\boxed{ \tt{ \: amp(z) = {\bigg(e\bigg)}^{ - \: \dfrac{\pi}{2}} \: }}$