f(x) = Σ (sin( k1*k2*x)) / k1, k1=1 to 2000 , k2=2 . A Fourier series, inspired by http://www.Mathe-Kalender 2019, Mai ; Prof. E.Wegert, G.Semmler TU-Freiberg and Pamela Gorkin and U.Daepp Univers. Lewisburg, there cited Hakon Grönwall ( Math. Annalen 1912 p.228 ) , Lit(4). (Gibbssches Phänomen) . The grey area has the area of F = π*π; The math. integral f(x) is : integral Σ (-cos(k1*k2*x) / k1*k1*k2, k1 = 1 to n; For f(x)=y we used a vector built by the Matlab/Octave-command linspace (0, 2*π, yres) from 0 to 2*π ; The numerical integral results to 19,7392, this is up to 6 digits exactly 2*π*π, starting y-values must not be zero ! This integral is invariant to natural numbers k1 and k2 . Within 2*pi the number of ’steps‘ is – k2 -. The integrals of the sin-terms from 0 – 2pi vanishes and it is our opinion that in form of the ’stair‘ the ‚Gibbs Phänomen‘ Σ (sin(k*x)) >π/2, k=1 to n . is not seen in the integral because the ‚overswinging‘ near zero is compensated by ‚underswinging‘ near pi , but it remains in the limits 0 – pi. Many other Fourier Series were presented in ‚Fun with Fourier Series‘ by Robert Baillie , Math. CA 18.07.2017 , (Lit. 5) .
