I am trying to find a periodic function, similar to a sine wave, such that the minimum points won’t be right in the middle between the neighboring maximum points, and preferably with a parameter that would determine how close a minimum point is to the nearest maximum point.
I have found that the generalized Clausen function satisfies that requirement, but was wondering if anyone knows of a simpler function, which may be less computationally expensive. I need the function to be a single continuous function, and one that can be integrated, if possible.
How about shifting a parabola to the left and making it periodic? I.e., take $f$ where $$f(x)=\begin{cases} \hfil (x-a)^2 & \text{if $0 \leqslant x \leqslant 1$}\\ \hfil f(x-1) & \text{if $x>1$}\\ \hfil f(x+1) & \text{otherwise}? \end{cases}$$
If $a=\tfrac15$ you get
and $a=\frac13$ gets you
Do you require it to be continuous/differentiable or something?
Edit: The function I suggested is integrable but not continuous. If you want it continuous, we can take the slight modified version:
$$f(x) = \begin{cases} \frac{1}{a^2}(x-a)^2 & \text{if $0\leqslant x\leqslant a$}\\ \frac{1}{(a-1)^2}(x-a)^2 & \text{if $a\leqslant x \leqslant 1$}\\ f(x-1) & \text{if $x\geqslant 1$}\\ f(x+1) & \text{otherwise}. \end{cases}$$
where $0<a<1$. The illustration below shows what happens for different values of $a$.