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:

 $\sin(z+\pi)\;=\;-\sin{z},\qquad\cos(z+\pi)\;=\;-\cos{z}$

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):

 $e^{z+i\pi}\;=\;e^{z}e^{i\pi}\;=\;-e^{z}$
Title antiperiodic function AntiperiodicFunction 2015-12-16 15:19:14 2015-12-16 15:19:14 pahio (2872) pahio (2872) 12 pahio (2872) Definition msc 30A99 PeriodicFunctions QuasiperiodicFunction antiperiodicity antiperiodic antiperiod