= Vanishing harmonic moments of a polynomial conformal disc map
{title2=$M_n=0\quad(n\geq N)$}
If a degree-$N$ <polynomial> $f(\zeta)=\sum_{j=1}^Na_j\zeta^j$ maps the <unit disc> conformally onto $D$, <Green's theorem> expresses $M_n=\iint_D z^n\,dx\,dy$ as the residue of $f^n f'\sum_j\bar a_j\zeta^{-j}$ times $\pi$. Its lowest possible power is $n-N$, so no residue occurs when $n\geq N$. For a quadratic map, $M_0=\pi(|a_1|^2+2|a_2|^2)$ and $M_1=\pi a_1^2\bar a_2$. The first moment can be complex; the zeroth is area.
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