Lemma: Analyticity and discontinuity

The functions $F_i$ are analytic off the cut, real for real $z<a$, and satisfy $F_i(0)=0$. For $s>a$,

Their real boundary values are continuous at threshold and grow at most logarithmically at infinity.

Proof. The densities are bounded. On compact subsets off the cut, the subtracted integrand is $O(u^{-2})$, uniformly with its required $z$ derivatives. This proves convergence and analyticity. Reality and $F_i(0)=0$ follow directly.

The distributional identity

gives (eq:disc). Near threshold, the densities vanish at least as $(u-a{)}^{1/2}$, so their Cauchy transforms have finite continuous boundary values there. The bounded limiting densities at large $u$ give logarithmic growth. Explicit formulas below also establish these boundary statements directly.