= Solution
For <complex differentiability at a point>, the derivative must have the same value along real and imaginary increments. Along real increments it equals $u_x+iv_x$, while along imaginary increments it equals $(u_y+iv_y)/i=v_y-iu_y$. Equating real and imaginary parts gives the <Cauchy-Riemann equations>
$$
\boxed{u_x=v_y,\qquad u_y=-v_x}.
$$
For the given real part,
$$
u_x=\frac{y^2-x^2}{(x^2+y^2)^2},\qquad u_y=\frac{-2xy}{(x^2+y^2)^2}.
$$
Integrating the <Cauchy-Riemann equations> gives a <harmonic conjugate>
$$
\boxed{v(x,y)=-\frac{y}{x^2+y^2}+C,\quad F(z)=\frac1z+iC,\quad \mathcal D=\mathbb C\setminus\{0\}},\qquad C\in\mathbb R.
$$
This <open set> is open and connected; $F$ is <holomorphic> there with derivative $-z^{-2}$. The origin must be excluded because the real part is undefined there. On this connected <open set>, any other <harmonic conjugate> differs by a real constant: the <Cauchy-Riemann equations> make both partial derivatives of the difference zero.
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