Is $\operatorname{erfc}(a + b\cos \theta)\operatorname{erfc}(c + d\sin \theta)$ periodic?

Question

$\DeclareMathOperator{\erfc}{erfc}$ I know that, $\erfc(\cos(\theta))\erfc( \sin(\theta))$ is periodic within the interval $\theta \in [0, 2\pi]$.

Can I show the periodic behaviors of $\erfc(a + b\cos(\theta))\erfc(c + d\sin(\theta))$

Where, $a,b,c,d \in \Bbb R$.

I plot that in Matlab, and it seems periodic. But how can I prove this analytically (Any hint will be appreciated)?

Answer 1

Any well defined dunction odf a periodic function is a periodic function. So is in this case and the period of this function is $2\pi$, as $f(x+2\pi)=f(x)$.

Answer 2

$f(\theta)=\text{erfc}(a+b\cos(\theta))\text{erfc}(c+d\sin(\theta))$ is periodic with period of $2\pi$ because $\sin$ and $\cos$ are. Indeed:

$$\begin{array}{rcl}f(\theta+2\pi)&=&\text{erfc}(a+b\cos(\theta+2\pi))\text{erfc}(c+d\sin(\theta+2\pi))\\&=&\text{erfc}(a+b\cos(\theta))\text{erfc}(c+d\sin(\theta))\\&=&f(\theta)\end{array}$$

just because $\cos(\theta+2\pi)=\cos\theta$ and $\sin(\theta+2\pi)=\sin\theta$.