If a <holomorphic function> $g$ on the <punctured complex plane> satisfied $e^{g(z)}=z$, the <chain rule> would give $e^{g(z)}g'(z)=1$, hence $g'(z)=1/z$. This contradicts the previous part. Therefore \b[there is no global holomorphic logarithm on the punctured plane]. A <branch of the complex logarithm> requires a suitable smaller domain.

Solution

Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2005 / 4 / iv / Solution

 Solution (source code)