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From: Geoff on 3 May 2010 17:32
I am currently reading the proof of Proposition 20 on P130 of the
book. He defines g(z) = ( eta(z) eta(2z) )^8 and I understand that
its q-expansion about infinity is q \prod (1 - q^n)^8 (1 - q^{2n})^8
so that g vanishes at infinity. Now, the sentence I do not get, which
may be very easy:

"Using the relation eta(-1/z) = sqrt(z/i) eta(z), we easily see that
g(z) vanishes at the cusp 0 as well."

Just to be sure, I understand the relation is just the functional
equation for the eta function and I have already gone through the
proof of that. I am just not understanding how to use this relation
to show that g vanishes at 0 as well.

Thanks
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