Adding a z coordinate of q produces a surface resembling a squat, blunt-nosed bomb with four infinitely spreading tailfins, volume = 4.231765651557.
Actually, it looks rather hypersonic if you crop the infinite tail and stretch it a bunch: http://gosper.org/sst.png .
volume = 4.231765651557. Per Macsyma. Mma is still working on the twelve digits I asked for yesterday.
This exercise exposed many deficiencies in my Macsyma Theta function package
Mostly in the area of simplifying special values and derivatives. Analogous to trig (under control of the same (%piargs) switch), it now translates out quarterperiods and quarterquasiperiods (which can get messy): (c430) (block([%piargs : false],thetaderiv[1](z+%pi/2+%i*log(q)/2,3,q,1)), %% = radcan(subst([q = 1/q,q = 1/q],%%))) %i log(q) %pi (d430) thetaderiv (z + --------- + ---, 3, q, 1) = 1 2 2 %i z %i z - (%i %e theta (z, q) + %e thetaderiv (z, 5, q) 3 3 %i z %i z + 5 %i %e thetaderiv (z, 4, q) - 10 %e thetaderiv (z, 3, q) 3 3 %i z %i z 5/4 - 10 %i %e thetaderiv (z, 2, q) + 5 %e thetaderiv (z, 1, q))/(4 q ) 3 3 Note that the rhs is free of d/dq. (c431) taylor(exponentialize(makeseries(%)),q,0,6) (d431)/T/ %i z %i z 4 %i z 8 11/4 %e %i (243 (%e ) - 1) %i ((3125 (%e ) - 243) %i) q - --------- - ---------------------- - --------------------------------- +... 5/4 %i z 1/4 %i z 3 4 q (4 %e ) q 4 (%e ) %i z %i z 4 %i z 8 11/4 %e %i (243 (%e ) - 1) %i ((3125 (%e ) - 243) %i) q = - --------- - ---------------------- - --------------------------------- +... 5/4 %i z 1/4 %i z 3 4 q (4 %e ) q 4 (%e ) The peculiar subst([q=1/q,q=1/q],...) is a sneaky way of temporarily asserting assume(not equal(q,0)). The special values of the eighth-periods of theta': (c407) for s thru 4 do block([%piargs : false,fancy_display : false], thetaderiv[s](%pi/4,1,q),print(%% = (if %piargs : true then resimplify(%%))))$ 3 2 %pi sqrt(2) eta (- q ) thetaderiv (---, 1, q) = - ------------------ 1 4 1/4 %i 3 2 %pi sqrt(2) eta (- q ) thetaderiv (---, 1, q) = ------------------ 2 4 1/4 %i %pi 3 8 thetaderiv (---, 1, q) = - 4 eta (q ) 3 4 %pi 3 8 thetaderiv (---, 1, q) = 4 eta (q ) 4 4 The 9th derivative wrt q at q=0 of the nth derivative wrt z of theta[3](z,q): (c436) thetaderiv[3](z,n,0,9) n %pi n (d436) 725760 6 cos(6 z + -----) 2 I wonder if WRI would like one of these.-) --rwg> --rwg