Dear friends,

I will present a series of commands for you to ponder that should speak for themselves.

> q := n -> int(1/(1-2^n*exp(I*theta*n))^2*2*I*exp(I*theta), theta=0..2*Pi);
                            2 Pi
                           /
                          |          2 I exp(theta I)
               q := n ->  |      ------------------------ dtheta
                          |            n                2
                         /       (1 - 2  exp(theta n I))
                           0

> seq(q(n), n=1..6);
                                        -8
                            0, 0, 0, 0, -- I Pi, 0
                                        25

> evalf(Int(1/(1-2^5*exp(I*theta*5))^2*2*I*exp(I*theta), theta=0..2*Pi));
memory used=5296.0MB, alloc=102.1MB, time=111.56
                                                    -24
                      0. - 0.18996217048363670877 10    I

> f := (n, x) -> 1/(1-x^n)^2;
                                              1
                           f := (n, x) -> ---------
                                                n 2
                                          (1 - x )

> z:= t -> 2*exp(I*t);
                             z := t -> 2 exp(t I)

> seq(abs(evalf(Int(f(n, z(t))*D(z)(t), t=0..2*Pi)/(2*Pi*I))), n=1..6);
memory used=5372.3MB, alloc=102.1MB, time=113.12
memory used=5448.6MB, alloc=102.1MB, time=114.65
memory used=5524.9MB, alloc=102.1MB, time=116.20
memory used=5601.2MB, alloc=102.1MB, time=117.73
memory used=5677.4MB, alloc=102.1MB, time=119.24
memory used=5753.7MB, alloc=102.1MB, time=120.79
                         -22                           -23
0.19674550473645331505 10   , 0.31010301501460449640 10   ,

                             -24                           -24
    0.57751607988237238365 10   , 0.11466915621134072267 10   ,

                             -25                           -26
    0.30952918217357836475 10   , 0.70632340779213408356 10

Would anyone care to comment? Read more here.

Marko Riedel