antiperiodic function

A special case of the quasiperiodicity (http://planetmath.org/Period3) 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:

$$\sin (z+\pi )=-\sin z,\cos (z+\pi )=-\cos z$$

The tangent (http://planetmath.org/Trigonometry) 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):

$$e^{z+i\pi }=e^z{e}^{i\pi }=-e^z$$