For odd d, 'limit(theta[3](0,%e^(2*%i*%pi*n/d)*r)*sqrt(-log(r)),r,1) = sqrt(%pi/d)*%e^(%i*%pi*(-2*g(4*n/d)+5*g(2*n/d)-2*g(n/d))/(12*d)) 2 i pi n -------- d limit theta (0, e r) sqrt(- log(r)) = r -> 1 3 4 n 2 n i pi (- 2 g(---) + 5 g(---) - 2 g(n/d)) d d --------------------------------------- pi 12 d sqrt(--) e d Where, as with the eta limits, g(r) := if integerp(r) then r else (floor(r),denom(r)*(%%+3)-(g(1/(r-%%))+1/denom(r))/(r-%%)) g(r) := if integerp(r) then r else (floor(r), 1 1 g(------) + -------- r - %% denom(r) denom(r) (%% + 3) - --------------------), r - %% a n(ot obviously) integer-valued function. The exponent expression -2*g(4*n/d)+5*g(2*n/d)-2*g(n/d) 4 n 2 n n - 2 g(---) + 5 g(---) - 2 g(-) d d d appears to be 0 mod 6, period n for fixed n, period d for fixed d, but otherwise no simpler than g. Even d looks not much harder. (Betcha you reparsed!) --rwg PS, how can a natural boundary grow only like 1/sqrt(-log q)? The other thetas are *much* wilder.