= Solution
Insert the specified real part into the polar <Cauchy-Riemann equations>. The first gives $v_r=nr^{n-1}\sin(n\theta)$, and integration with respect to $r$ gives
$$
v(r,\theta)=r^n\sin(n\theta)+C(\theta).
$$
The second equation then requires $v_\theta=nr^n\cos(n\theta)$, so $C'(\theta)=0$. The remaining imaginary additive constant is fixed by $f(0)=0$. Thus
$$
\boxed{v(r,\theta)=r^n\sin(n\theta),\qquad f(z)=r^ne^{in\theta}=z^n.}
$$
The resulting <polynomial> is holomorphic at the origin too, consistently extending the calculation made in polar coordinates with $r>0$.
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