Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 31E / Solution

Orient the unit circle counterclockwise and use the standard principal-value Cauchy operatorFor a Fourier/Laurent mode $t^n$ it gives $+t^n$ when $n\ge 0$ and $-t^n$ when $n<0$. This follows from the interior/exterior Cauchy projections, or by direct indentation of the pole. Decompose $\varphi ={\varphi }_{+}+{\varphi }_{-}$ into its nonnegative and negative modes, so $K\varphi ={\varphi }_{+}-{\varphi }_{-}$. LetThe printed equation reduces toPositive-power coefficients force ${\varphi }_{+}=J/2$, a constant; powers at most $-2$ force ${\varphi }_{-}=0$. The $t^{-1}$ coefficient then gives $-J=2$, and the integral definition is consistent with $c_0=-1$. ThusDirect substitution has $K\varphi =-1$, $J=-2$ and gives $2t^{-1}$ as required. The same projection argument on the homogeneous equation forces $J=0$ and both components zero, proving uniqueness in the usual Hölder or $L^2$ class for this singular-integral problem.