TITLE: Are we able to "combine" two analytic continuation to get another?
QUESTION [1 upvotes]: Say, I want to find the analytic continuation of the function $f(z)$ defined on $D=\{a:-\pi/5<arg(a)<\pi/5\}$ to the slit plane $\mathbb{C}-(-\infty,0]$. I assume that $f(z)$ is real on the boundary of $D$ and has analytic continuation on $\bar D$. I can shift $f(z)$ to $f(ze^{-\pi/5})$ and do the Schwarz reflection principle. I can do this twice, shift it back, and restrict the domain to get the analytic continuation on the upper half plane. I can do the same thing on the lower half plane to get the analytic continuation on the lower half plane. But I don't know if I'm able to combine these two analytic continuations to get the analytic continuation on the whole slit plane. Can anybody give me any tips?..

REPLY [1 votes]: If $f$ is analytic for $arg(z)\in (a,b),z\ne 0$ continuous on $arg(z) \in [a,b],z\ne 0$ and real on $arg(z)=b,z\ne 0$ then the Schwartz reflection along $arg(z)=b$ gives that $f$ extends analytically to $arg(z)\in (a,b+(b-a))$, and it is continuous on $arg(z)\in [a,b+(b-a)]$ and real on $arg(z)=2b-a$. So we can repeat and extend more and more the domain.
If $2b-2a>2\pi$ then $f(e^{2i\pi}z)$ doesn't have to be the same as $f(z)$, our analytic continuation may be "multi-valued" (try with $f(z)=z^{\pi},a=0,b=2$) that's the point of your branch cut on $(-\infty,0]$.
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