Multivariable periodic function?

Question

Suppose I have the function:

$u(t,x)=sin(x)cos(2t)+cos(x)sin(t)cos(t)$

How can I know if this function is a periodic function of $t$ if it includes two variables?

Answer 1

$u(t+2\pi,x) = \sin(x) \cos(2(t+2\pi)) + \cos(x) \sin(t + 2\pi) \cos(t + 2 \pi) = \sin(x) \cos(2t) + \cos(x) \sin(t) \cos(t) = u(t,x)$

In fact we have $u(t+\pi,x) = u(t,x)$.

Answer 2

I totally agree with the accepted answer. I just wanted to add something more general. It may be obvious, but because I couldn't find it somewhere, I thought perhaps it may help someone in the future.

Consider a scalar multivariable function $f(x_1,x_2,...,x_n)$.

• We could say that f is periodic with period $T_k$ with respect to $x_k$ for some $k \in \{1,2,...,n\}$ if and only if $$f(x_1,...,x_k + \lambda T_k,...,x_n) = f(x_1,...,x_k,...,x_n) \text{, } \forall \lambda \in \mathbb{Z}$$

Note that $f$ could be for example: $T_1$-periodic with respect to $x_1$ and $T_5$-periodic with respect to $x_5$, with $T_1$ and $T_5$ different in general.

• From my experience(*), when someone says that $f$ is, for example, $2\pi$-periodic, they mean that f is $2\pi$-periodic with respect to $x_1$ and with respect to $x_2$ and $...$ and with respect to $x_n$.

(*) E.g. in the book Higher-Order Spectra Analysis: A Nonlinear Signal Processing Framework , regarding cumulant spectra.