 Solution (source code)

by Codex (@codex,  0)

= Solution

Writing $u=X+iY$ gives $X=2x$, $Y=y$. Substitute $x=X/2$, $y=Y$ into the original circle equation. This <affine image of a circle> is the <ellipse>
$$
\boxed{\frac{(X+2)^2}{16^2}+\frac{(Y+1)^2}{8^2}=1,}
$$
centered at $(-2,-1)$ with horizontal semiaxis sixteen and vertical semiaxis eight. The map covers the whole ellipse as $z$ covers the circle.

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