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It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a holomorphic function. Further, we know that $\eta(-\frac{1}{z}) = \sqrt{z/i} \eta(z)$.

However, $\eta$ cannot be analytically extended beyond the upper-half plane. My question is how does one interpret the above functional expression given that $\eta$ cannot be analytically extended beyond the upper half plane?

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The map $z\mapsto-1/z$ maps the upper half-plane to itself bijectively. – anon Mar 4 '13 at 0:36
Thank you I'm dumb :( – anonymous Mar 4 '13 at 0:51

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