Looking for examples of functions that are analytic only in the strip $(0,1)$ and with natural boundaries $\Re(z)=0,1.$
I thought of modular forms but they only have one natural boundary I think.
2026-05-05 13:41:40.1777988500
examples of functions analytic only in strip (0,1) and with natural boundaries Re(z)=0,1?
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Take a known example with natural boundary. Say $f(z) = \sum_{n=0}^\infty z^{2^n}$ with natural boundary $|z|=1$. Then use a conformal mapping $\varphi$ that maps your strip onto $|z| < 1$. Then $f(\varphi(z))$ is your answer.