Show that the function $f (z) = 1 / z$ transforms a circle centered at the origin in the $xy$ plane in a circle centered at the origin in the $uv$ plane. Someone can give me a clue to start the demostration, I have the doubt if I separate function in its real and imaginary parts and independently plot.
I try to separate the real and imaginary part of the funcion then:
$z=\frac{1}{x+iy}=\frac{i}{x+iy}*\frac{x-iy}{x-iy}=\frac{x-iy}{x^2+y^2}$ then $u=\frac{x}{x^2+y^2}$ and $v=\frac{-y}{x^2+y^2}$ How can this help me??
Best Answer
Hint: $\;z\in\Bbb C\;$ is on a circle centered at the origin, say of radius $\;r\;$, iff $\;|z|=r\;$
Well, now check what happens to the above $\;z\;$ under $\;f(z):=\frac1z\;$ ...