Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 1 / d / Solution

The same Volterra series converges uniformly on the entire compact characteristic square $0\le \xi ,\eta \le 1$, becauseThe boundary functions are analytic on neighbourhoods of the compact axis segments, so finitely many complex neighbourhoods give uniform Cauchy estimates for their derivatives. Applying $T^m$ adds the two factorial denominators above, and the corresponding derivative series converges on a neighbourhood of every point of the closed square. Thus the local analytic solutions continue across the whole square and agree on overlaps by uniqueness.

Equivalently, the integral equation bounds $u$ and every differentiated equation on each smaller rectangle; no norm can blow up at a first missing corner. The local analytic existence theorem therefore extends the solution through that corner. This is Global continuation for the analytic Goursat problem.