Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2007 / B / 6 / a / Solution

Conformality implies $f^{\prime }(0)\ne 0$, so $a_1\ne 0$. Apply the stated Green's theorem with $F=z^n\bar{z}$:Here $\bar{\zeta }={\zeta }^{-1}$ on the unit circle. The lowest power in $f^n{f}^{\prime }$ is ${\zeta }^n$, while the most negative power in the conjugate polynomial is ${\zeta }^{-N}$. For $n\ge N$ their product has no ${\zeta }^{-1}$ term. The residue theorem gives $M_n=0$ for all $n\ge N$.

When $N=2$, the residue for $n=0$ is $|a_1{|}^2+2|a_2{|}^2$. For $n=1$, expandOnly $a_1^2\zeta {\bar{a}}_2{\zeta }^{-2}$ contributes. ThereforeThe first is the area; the second records its complex first moment, and need not be real.