= Solution
The same Volterra series converges uniformly on the entire compact characteristic square $0\leq\xi,\eta\leq1$, because
$$
\sum_{m\geq0}\frac{1}{4^m(m!)^2}<\infty.
$$
The 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>.
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