antiperiodic

An antiperiodic function or sequence is one that reverses sign after a fixed shift. For a function f on the real line, antiperiodicity with offset T is expressed by f(x + T) = −f(x) for all x. This relation implies f is periodic with period 2T, since f(x + 2T) = f(x), but T itself is not a period in the usual sense and is often called an antiperiod.

Simple examples include trigonometric and complex exponentials: sin(x) and cos(x) satisfy f(x + π) = −f(x), so they are antiperiodic with T = π (and periodic with period 2π). More generally, e^{ix} is antiperiodic with T = π. For discrete sequences, a_n is antiperiodic with shift N if a_{n+N} = −a_n, which implies periodicity with period 2N.

Antiperiodic conditions influence spectral and harmonic structure. When expanded as a Fourier series over interval length 2T, an antiperiodic function contains only odd harmonics relative to that fundamental interval, corresponding to frequencies (2k+1)π/T. In mathematical physics and geometry, antiperiodic (or twisted) boundary conditions appear in studies of fermions at finite temperature, where fields are antiperiodic in imaginary time, and in problems on nonorientable manifolds like the Möbius strip. They are also used in the analysis of differential operators with sign-reversing boundary conditions, producing characteristic changes in eigenvalue distributions and mode symmetries.