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 January 5th, 2014, 04:13 AM #1 Newbie   Joined: Jan 2014 Posts: 2 Thanks: 0 analytic extension Show that there is no analytic extension of $logx , x>0$ on $C\backslash \{ 0 \}$
 January 26th, 2014, 01:44 PM #2 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions Re: analytic extension Assume that $\mathrm{Log}(z)$ is analytic on the domain $\mathbb{C} \backslash \, \{0\}$ then it's derivative $\frac{1}{z}$ must be also analytic on the same domain which implies that any closed curve around the origin must vanish. Since $\int_{|z|=1} \frac{1}{z}\, dz = 2\pi i$ , we get a contradiction.

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