= Homogeneous freedom in a scalar singular integral equation
{title2=$w=\widetilde w(C\rho+H)$}
For the scalar jump $(a+b)w_++(b-a)w_-=c$, divide by a nonvanishing canonical homogeneous solution $\widetilde w$ to obtain an additive jump $\rho=c/((a+b)\widetilde w_+)$. Its <Cauchy integral> gives a particular quotient, while every additional quotient $H$ with no jump is homogeneous freedom. The density $f=w_+-w_-$ therefore differs from the normalized particular density by $-2b\widetilde w_+H/(a-b)$. Normalization, index and endpoint conditions determine the allowable $H$; dropping it without such conditions does not give the general solution.
Back to article page
