The two holomorphic function functions agree throughout their overlap region strip, so they define a single holomorphic function on the union of their domains, which is the whole complex plane. The resulting entire function is bounded outside a sufficiently large disk by the assumptions at infinity. It is bounded on the disk by continuity and compactness. Liouville theorem therefore makes it constant:
(1)
Agreement on an open overlap region is what permits analytic continuation; equality of unspecified boundary values alone would need additional hypotheses.

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