Enormous screwing around gave eta(%i*q)=(%e)^(-((5*%i*%pi)/48))*eta(q^4) *(((4*eta(q^8)^2)/(eta(q^2)^2))+((%i*eta(q^2)^2)/(eta(q^8)^2)))^(1/4)
5 %i %pi - -------- 2 8 2 2 48 4 4 eta (q ) %i eta (q ) 1/4 eta(%i q) = %e eta(q ) (---------- + -----------) , 2 2 2 8 eta (q ) eta (q )
valid in the wedge -pi/8 < arg(q) <= pi/8.
Ouch, no, a complicated subset of that. Something has gnawed holes at the outer edge of the pizza wedge extending in almost to |q|=.8 . Near |q|=1 there even appear to be holes in the holes where equality reappears. Wow, it looks like the cross section of a cabbage head (or clover bud). (http://gosper.org/cabbage.gif) At least it includes positive q. --rwg (Six or nine tries with firefox never loaded the entire image. Chrome loaded it all first try, but then clipped off the top!)
Not clover. The branches branch. http://gosper.org/cabbage.png . IWBN to know if that ganglion is > single-point connected to its stem and fronds. And the voids branch so that the level set isn't simply connected. (Wider view: http://gosper.org/cabbage2.png . Left edge is -pi/7, showing expected radial fault at -pi/8.) --rwg