In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function f {\displaystyle f} is quasiperiodic with quasiperiod ω {\displaystyle \omega } if f ( z + ω ) = g ( z , f ( z ) ) {\displaystyle f(z+\omega )=g(z,f(z))} , where g {\displaystyle g} is a "simpler" function than f {\displaystyle f} . What it means to be "simpler" is vague.

A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:

f ( z + ω ) = f ( z ) + C {\displaystyle f(z+\omega )=f(z)+C}

Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:

f ( z + ω ) = C f ( z ) {\displaystyle f(z+\omega )=Cf(z)}

An example of this is the Jacobi theta function, where

ϑ ( z + τ ; τ ) = e − 2 π i z − π i τ ϑ ( z ; τ ) , {\displaystyle \vartheta (z+\tau ;\tau )=e^{-2\pi iz-\pi i\tau }\vartheta (z;\tau ),}

shows that for fixed τ {\displaystyle \tau } it has quasiperiod τ {\displaystyle \tau } ; it also is periodic with period one. Another example is provided by the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Bloch's theorem says that the eigenfunctions of a periodic Schrödinger equation (or other periodic linear equations) can be found in quasiperiodic form, and a related form of quasi-periodic solution for periodic linear differential equations is expressed by Floquet theory.

Functions with an additive functional equation

f ( z + ω ) = f ( z ) + a z + b   {\displaystyle f(z+\omega )=f(z)+az+b\ }

are also called quasiperiodic. An example of this is the Weierstrass zeta function, where

ζ ( z + ω , Λ ) = ζ ( z , Λ ) + η ( ω , Λ )   {\displaystyle \zeta (z+\omega ,\Lambda )=\zeta (z,\Lambda )+\eta (\omega ,\Lambda )\ }

for a z-independent η when ω is a period of the corresponding Weierstrass ℘ function.

In the special case where f ( z + ω ) = f ( z ) {\displaystyle f(z+\omega )=f(z)} we say f is periodic with period ω in the period lattice Λ {\displaystyle \Lambda } .