> restart;
> a:=(k,p)->exp(2*I*p*Pi/k);

                                        2 I p Pi
                     a := (k, p) -> exp(--------)
                                           k

> f:=(p,k,t)->(1-t)*a(k,p)+t*a(k,p+1);

          f := (p, k, t) -> (1 - t) a(k, p) + t a(k, p + 1)

> g:=(k,t)->(1-k*t+floor(k*t))*a(k,floor(k*t))+(k*t-floor(k*t))*a(k,floor(k*t)+1);

  g := (k, t) -> (1 - k t + floor(k t)) a(k, floor(k t))

         + (k t - floor(k t)) a(k, floor(k t) + 1)

> with(plots);

  [animate, animate3d, animatecurve, arrow, changecoords, complexplot,

        complexplot3d, conformal, conformal3d, contourplot,

        contourplot3d, coordplot, coordplot3d, densityplot, display,

        dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d,

        graphplot3d, implicitplot, implicitplot3d, inequal,

        interactive, interactiveparams, intersectplot, listcontplot,

        listcontplot3d, listdensityplot, listplot, listplot3d,

        loglogplot, logplot, matrixplot, multiple, odeplot, pareto,

        plotcompare, pointplot, pointplot3d, polarplot, polygonplot,

        polygonplot3d, polyhedra_supported, polyhedraplot, rootlocus,

        semilogplot, setcolors, setoptions, setoptions3d, spacecurve,

        sparsematrixplot, surfdata, textplot, textplot3d, tubeplot]

> complexplot(g(3,t),t=0..1.4);

> c:=(k,n)->int(g(k,t/2/Pi)*exp(-I*n*t),t=0..2*Pi)/2/Pi;

                        /        2 Pi                          \
                        |       /                              |
                        | 1    |            t                  |
     c := (k, n) -> 1/2 |----  |      g(k, ----) exp(-I n t) dt|
                        | Pi   |           2 Pi                |
                        |     /                                |
                        \       0                              /

> c(3,2);

                                  0

> S:=(k,n,t)->sum('c(k,p)*exp(I*p*t)','p'=-n..n);

                                n
                              -----
                               \
           S := (k, n, t) ->    )     'c(k, p) exp(p t I)'
                               /
                              -----
                             'p' = -n

> S(3,4,0.5);

                    0.6747869737 + 0.2228812686 I

> complexplot([g(3,t/2/Pi),S(3,1,t),S(3,3,t),S(3,5,t)],t=0..2*Pi);

>