O17-085
> restart;
> f:=x->int(sin(x*t)/(exp(t)-1),t=0..infinity);

                              infinity
                             /
                            |           sin(x t)
                 f := x ->  |          ---------- dt
                            |          exp(t) - 1
                           /
                             0

> plot(f,-10..10);

> g:=cosh(t*x);

                            g := cosh(t x)

> c:=n->int(g*exp(-I*n*t),t=-Pi..Pi)/2/Pi;

                           /        Pi                 \
                           |       /                   |
                           | 1    |                    |
             c := n -> 1/2 |----  |    g exp(-I n t) dt|
                           | Pi   |                    |
                           |     /                     |
                           \       -Pi                 /

> assume(n,integer);factor(c(n)*exp(I*n*Pi));

                                                     n~ 2
                 x (exp(2 x Pi) - 1) exp(-x Pi) ((-1)  )
             1/2 ----------------------------------------
                                2     2
                              (x  + n~ ) Pi

g~(Pi)=2*(sinh(x*Pi)/Pi)*f(x)+c(0)=cosh(Pi*x)