kocosinus

kocosinus is a mathematical function that generalizes the standard cosine function by incorporating a scaling parameter k into its argument. Defined as kocosinus(k, x)=cos(kx), it allows the adjustment of the frequency of oscillation while preserving the amplitude and symmetry properties of the cosine waveform. The notation is often encountered in the analysis of periodic phenomena where a variable frequency component is required, such as in Fourier series representations of signals with modulated bandwidth. The function retains many classical identities of the cosine, including even symmetry, periodicity with period 2π/k, and the relation cos(kx)=Re(e^{ikx}).

In signal processing, the kocosinus function is used to construct band-limited waveforms that can be smoothly frequency-shifted by varying k. This property is exploited in chirp signal design, where the instantaneous frequency varies linearly with time. In applied physics, the kocosinus serves as an analytical tool for describing vibrational modes in systems with a variable stiffness parameter, as the mode shape can be expressed as cos(ωt) with ω depending on a system constant k. In mathematics, the function appears in studies of Bessel functions and spherical harmonics, where the radial part of solutions often contains cos(kx) terms.

The term "kocosinus" first appeared in the early 1970s in the context of non-homogeneous wave equations. Its application has since expanded to include digital filter design, telecommunications (frequency-hopping spread spectrum), and computational modeling of oscillatory systems. Because of its straightforward extension of the cosine, the kocosinus retains popularity as a pedagogical tool when introducing variable-frequency Fourier analysis.