MHB Help Me Solve $\text{Im}(z^3)$ in Rectangular Form

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To find the imaginary part of \( z^3 \) in rectangular form, where \( z = x + jy \), the expression simplifies to \( \text{Im}(x^3 + 3x^2jy - 3xy^2 - jy^3) \). The relevant terms containing \( j \) are \( 3x^2jy - jy^3 \), leading to \( \text{Im}((x^3 - 3xy^2) + (3x^2y - y^3)j) \). The imaginary part is identified as \( 3x^2y - y^3 \). Thus, the final answer for \( \text{Im}(z^3) \) is \( 3x^2y - y^3 \).
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I don't how to proceed further

Let $z=x+jy$ find in rectangular form:

1. $\text{Im}(z^3)$

this is where I can get to,

$\text{Im}(x^3+3x^2jy+3xj^2y^2+j^3y^3)=\text{Im}(x^3+3x^2jy-3xy^2-jy^3)$

please help me.
 
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Drain Brain said:
I don't how to proceed further

Let $z=x+jy$ find in rectangular form:

1. $\text{Im}(z^3)$

this is where I can get to,

$\text{Im}(x^3+3x^2jy+3xj^2y^2+j^3y^3)=\text{Im}(x^3+3x^2jy-3xy^2-jy^3)$

please help me.

So what are the terms which have a "j" part?
 
Drain Brain said:
I don't how to proceed further

Let $z=x+jy$ find in rectangular form:

1. $\text{Im}(z^3)$

this is where I can get to,

$\text{Im}(x^3+3x^2jy+3xj^2y^2+j^3y^3)=\text{Im}(x^3+3x^2jy-3xy^2-jy^3)$

please help me.

Hi Drain Brain! :)

You're already there:
$$\text{Im}(x^3+3x^2jy-3xy^2-jy^3) = \text{Im}((x^3-3xy^2)+(3x^2y-y^3)j)
= (3x^2y-y^3)$$
 
they are $3x^2yj-j^3$ is this the answer?
 
Drain Brain said:
they are $3x^2yj-j^3$ is this the answer?

no .as I like serena mentioned it is
$3x^2y- y^3$ because imaginary part of $x+jy$ is $y $
 
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