Anyway, here's that weird Abel-Plana integral in terms of theta(0) only:
inf / [ sin(y) (d188) I ------------------------------------- dy = ] 2 pi y / ------- 0 x (e - 1) (cos(2 y) + cosh(2 x))
2 2 pi pi - --- - ---- x 2 x 2 x - 4 acot(e ) - x (sech(x) + 1) + pi theta (0, i e ) theta (0, i e ) 3 4 -----------------------------------------------------------------------------. 8 sinh(x)
Testing for x=pi:
(c189) dfloat(opsubst(nounify(integrate)=lambda([[l]],apply(quad_inf,l)),subst(%pi,x,%)))
(d189) 0.00132907964796d0 = 0.00132907964796d0
I don't recall seeing elementary integrals for thetas. --rwg
Duh, http://en.wikipedia.org/wiki/Theta_function#Integral_representations Looks like they just construct the powerseries as an infinite sum of residues. I overestimated the comprehensiveness of W&W. New business: Can someone tell me if Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124, or some other source answers whether A050788 is known infinite, i.e., does c^3+1=a^3+b^3 infinitely often? And does ceiling((a^n+b^n)^(1/n))^n - a^n - b^n < k have only finitely many solutions for n>3? --rwg