Why is this function $2\pi n$ periodic?

Question

We have the $n$'th function defined as

\begin{equation} f_{n}(r,\theta)=\int_{0}^{2\pi}w(r,\theta,R,\theta')cos(n\theta')d\theta' \end{equation}

where $R$ is just some constant. I know that $w$ is $2\pi$ periodic in $\theta$ (and in $\theta'$ I think).

I can't work out how we know $f_{n}$ is $2\pi n$ periodic in $\theta$?

Any help would be appreciated. Thanks!

Answer 1

Note that $w$ is $2\pi$ periodic in $\theta$ and thus $2n\pi$ periodic in $\theta$, which implies that $$ \int_{0}^{2\pi}w(r,\theta+2n\pi,R,\theta')\cos(n\theta')\ d\theta'=\int_{0}^{2\pi}w(r,\theta,R,\theta')\cos(n\theta')\ d\theta' $$