Find $x$ and $y$ in terms of real and imaginary parts of $f(z)=\frac{1}{z}$

Question

Find $x$ and $y$ in terms of real and imaginary parts of $f(z)=\frac{1}{z}$.

Here is what I have tried so far: $$f(z)=\frac{1}{z}=\frac{1}{x+iy}=\frac{x-iy}{x^2+y^2}=u(x,y)+iv(x,y) \\ \implies u(x,y)=\frac{x}{x^2+y^2}, \quad v(x,y)=\frac{-y}{x^2+y^2}$$

Now, how do you find $x$ and $y$ in terms of $u$ and $v$?

I am stuck, I keep finding equalities like $x=x$. A hint would be welcome.

Answer 1

Since$$u(x,u)+v(x,y)i=\frac1{x+yi},$$you have$$x+yi=\frac1{u(x,u)+v(x,y)i}=\frac{u(x,y)}{u^2(x,y)+v^2(x,y)}-\frac{v(x,y)}{u^2(x,y)+v^2(x,y)}i.$$