A special case of the quasiperiodicity of functions is the antiperiodicity. An antiperiodic function satisfies for a certain constant the equation
for all values of the variable . The constant is the antiperiod of . Then, has also other antiperiods, e.g. , and generally with any .
The antiperiodic function is always as well periodic with period , since
Naturally, then there are all periods with .
Not all periodic functions are antiperiodic.
For example, the sine and cosine functions are antiperiodic with , which is their absolutely least antiperiod:
The tangent and cotangent functions are not antiperiodic although they are periodic (with the prime period ; see complex tangent and cotangent).
The exponential function is antiperiodic with the antiperiod (see Euler relation):