complex numbers - How to break $frac{1}{z^2}$ into real and imaginary parts?

$$ \frac{1}{(x+iy)^2}=\frac{1}{x^2+i2xy-y^2}=\frac{x^2}{(x^2+y^2)}-\frac{2ixy}{(x^2+y^2)}-\frac{y^2}{(x^2+y^2)}$$
So I thought I could just say:
$$ Re(\frac{1}{z^2})=\frac{x^2}{(x^2+y^2)}-\frac{y^2}{(x^2+y^2)}$$
and
$$ Im(\frac{1}{z^2})=-\frac{2ixy}{(x^2+y^2)}$$
But I know that is wrong because it looks nothing like the graph of the real part of 1/z^2 on wolfram alpha found here: http://www.wolframalpha.com/input/?i=1%2F(x%2Bi*y)%5E2

Then I thought I could must multiply $1/z^2$ by $z/z$ to get $\frac{x}{z^3}$ and $\frac{iy}{z^3}$ however graphing these again shows that they are not the real and complex parts of $\frac{1}{z^2}$.

Answer

Notice, when $z\in\mathbb{C}$:

$$z=\Re[z]+\Im[z]i$$

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So, we get (in steps):

• $$z^2=\left(\Re[z]+\Im[z]i\right)^2=\Re^2[z]-\Im^2[z]+2\Re[z]\Im[z]i$$


• $$\overline{z^2}=\overline{\Re^2[z]-\Im^2[z]+2\Re[z]\Im[z]i}=\Re^2[z]-\Im^2[z]-2\Re[z]\Im[z]i$$


• $$z^2\cdot\overline{z^2}=|z|^4=\left(\sqrt{\Re^2[z]+\Im^2[z]}\right)^4=\left(\Re^2[z]+\Im^2[z]\right)^2$$

Now, we get:

$$\frac{1}{z^2}=\frac{\overline{z^2}}{z^2\cdot\overline{z^2}}=\frac{\Re^2[z]-\Im^2[z]-2\Re[z]\Im[z]i}{\left(\Re^2[z]+\Im^2[z]\right)^2}$$

So:

• $$\color{red}{\Re\left[\frac{1}{z^2}\right]=\frac{\Re^2[z]-\Im^2[z]}{\left(\Re^2[z]+\Im^2[z]\right)^2}}$$


• $$\color{red}{\Im\left[\frac{1}{z^2}\right]=-\frac{2\Re[z]\Im[z]}{\left(\Re^2[z]+\Im^2[z]\right)^2}}$$