= Simply connected corollary of the monodromy theorem
On a <simply connected domain> $D$, a <germ of a holomorphic function> admitting <analytic continuation> along every <path> from its base point determines a unique <holomorphic function> $F:D\to\mathbb C$ with that initial <germ of a holomorphic function>. <A Guide to Complex Variables by Steven G. Krantz>[Section 9.3.6, printed p. 159: globally defined holomorphic function]{ref}
Indeed, any two <paths> with the same endpoints are <homotopic> with endpoints fixed. Define $F(b)$ using the terminal <germ of a holomorphic function>. The <monodromy theorem> makes this independent of the <path>. In a small <open disc> about $b$, extend the chosen route by a short segment; the same local representative supplies $F$ throughout the <open disc>, proving <holomorphy>. The <identity theorem> proves uniqueness. The <existence hypothesis of the monodromy theorem> must still be checked.
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