[math-fun] theta_3(pi/3,exp(-pi/3))

8 Dec 2008


      I don't recall seeing special values of thetas at pi/3.  After a long
sequence of "miraculous" simplifications,

                  pi    - pi/3           pi    - pi/3
           theta (--, %e      ) = theta (--, %e      )
                3 3                    4 6

                                        1/8                         1
                                       3    sqrt(sqrt(3) - 1) Gamma(-)
                                                                    4
                                  =    -------------------------------.
                                                   1/4   3/4
                                                2 2    pi

rwg>  [...]  According to http://dlmf.nist.gov/2/10/
...
Sufficient conditions for the validity of this [Abel-Plana] result are:
...
1.(a) On the strip 0<=Re(z)<oo, h(z) is analytic in its
     interior, <the derivatives of h are> continuous on its
     closure, and h(z)=o(exp(2 pi Im(z)) as Im(z)->±oo, uniformly
     with respect to 0<=Re(z)<oo.
...
(b) h(z) is real when 0<=z<oo.
I should have punted (b).  I think it's only there because DLMF for some
reason wrote - 2 Im(h(i y)) instead of i (h(i y) - h(-i y)).
--rwg
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[math-fun] theta_3(pi/3,exp(-pi/3))

rwg＠sdf.lonestar.org