A special case of the quasiperiodicity of functions is the antiperiodicity. An antiperiodic function $f$ satisfies for a certain constant $p$ the equation $$f(z+p) = -f(z)$$ for all values of the variable $z$ . The constant $p$ is the antiperiod of $f$ . Then, $f$ has also other antiperiods, e.g. $-p$ , and generally $(2n\!+\!1)p$ with any $n \in \mathbb{Z}$ .

The antiperiodic function $f$ is always as well periodic with period $2p$ , since $$f(z+2p) = f((z+p)+p) = -f(z+p) = -(-f(z)) = f(z).$$ Naturally, then there are all periods $2np$ with $n \in \mathbb{Z}$ .

Not all periodic functions are antiperiodic.

For example, the sine and cosine functions are antiperiodic with $p = \pi$ , which is their absolutely least antiperiod: $$\sin(z+\pi) = -\sin{z}, \quad \cos(z+\pi) = -\cos{z}$$ The tangent and cotangent functions are not antiperiodic although they are periodic (with the prime period $\pi$ ; see complex tangent and cotangent).

The exponential function is antiperiodic with the antiperiod $i\pi$ (see Euler relation): $$e^{z+i\pi} = e^z e^{i\pi} = -e^z$$