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

Question

$$ \frac{1}{(x+iy)^2}=\frac{1}{x^2+i2xy-y^2}=\frac{x^2}{(x^2+y^2)}-\frac{2ixy}{(x^2+y^2)}-\frac{y^2}{(x^2+y^2)}$$ 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 1

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:

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

Answer 2

Your first proposition should be corrected by noting that when you multiply numerator and denominator by the conjugate of $x^2+2ixy-y^2$, the denominator becomes $(x+iy)^2(x-iy)^2=(x^2+y^2)^2$ and not $x^2+y^2$. Moreover, there's no $i$ in the imaginary part.

Answer 3

\begin{eqnarray} f(z)=1/(z^2)=1/((x+iy)^2)&=&1/(x^2-y^2+2ixy) \newline &=&(1/(x^2-y^2+2ixy))\cdot((x^2-y^2-2ixy)/(x^2- y^2-2ixy)) \newline &=&((x^2-y^2-2ixy)/((x^2-y^2)^2+4x^2y^2)) \newline &=&(x^2-y^2-2ixy)/(x^2+y^2)^2 \end{eqnarray} therefore $\Re(1/z^2)=(x^2-y^2)/(x^2+y^2)^2$ and $\Im(1/z^2)=-2xy/(x^2+y^2)^2.$