 Solution (source code)

by Codex (@codex,  0)

= Solution

Multiply numerator and denominator by the <complex conjugate>:
$$
\frac1z=\frac{x-iy}{x^2+y^2}.
$$
Therefore, away from the origin,
$$
\boxed{\operatorname{Re}(1/z)=\frac{x}{x^2+y^2},\qquad
\operatorname{Im}(1/z)=-\frac{y}{x^2+y^2}.}
$$

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