Functions sinKx and cosKx

The author introduces the piecewise linear functions sin K x and cos K x representing K straight line segments with vertices on sinx and cosx respectively. The author Fourier expands these functions, discusses their properties, and derives a number of identities which follow from the expansion of the functions themselves and their integrals or derivatives. The motivation to study sin K x and cos K x comes from the computer simulations of the Rayleigh-Taylor instability in which the eigenmodes sinx and cosx are represented by sin K x and cos K x, K+1 being the number of nodes used in the simulations. The author finds that the harmonics generated by a finite K representation occur only at multiples of K plus or minus one.