# Conditionally-periodic function

A function $A \circ \phi$ which is the composition of a $2 \pi$- periodic function $A: T ^ { n } \rightarrow \mathbf C$, where $T ^ { n }$ is the $n$- dimensional torus, and a function $\phi : \mathbf R \rightarrow \mathbf R ^ {n}$ such that $\dot \phi = \omega$, where $\omega = ( \omega _ {1} \dots \omega _ {n} )$ is a constant vector whose components are linearly independent over the rational numbers. Examples of conditionally-periodic functions are given by partial sums of Fourier series

$$\sum _ {i = 1 } ^ { n } [ A _ {i} \sin ( \omega _ {i} t + \psi _ {i} ) + B _ {i} \cos ( \omega _ {i} t + \psi _ {i} )],$$

where

$$A = A ( \phi _ {1} \dots \phi _ {n} ) = \ \sum _ {i = 1 } ^ { n } [ A _ {i} \sin \phi _ {i} + B _ {i} \cos \phi _ {i} ],$$

$$\phi = ( \phi _ {1} ( t) \dots \phi _ {n} ( t)) = ( \omega _ {1} t + \psi _ {1} \dots \omega _ {n} t + \psi _ {n} ).$$

If a conditionally-periodic function is continuous, then it coincides with a quasi-periodic function with periods $\omega _ {1} \dots \omega _ {n}$.

#### References

 [1] V.I. Arnol'd, "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR (1980) (Translated from Russian)