I am trying to solve the following functional equation, which appears in some of my physics calculations : $f(x)=\frac{1}{2}\left(f(\frac{x}{2})(1+\cos(\frac{x}{2}))+f(\frac{x}{2}+\pi)(1-\cos(\frac{x}{2}))\right)$,
for a function $f$ defined on $(0,2\pi)$.
I am interested in all functions satisfying this equation, but let us start by finding all functions which are defined on $[0,2\pi]$, which are continuous and differentiable.
Constant functions are solutions.
Let us note $f(0)=a$ and $f(2\pi)=b$. My numerics, starting from some function and iterating until I get to a fixed point seem to point towards a unique solution depending only on $a$ and $b$. If $a=b$, this would be the constant function.
If $a\neq b$, the solution seems to have the following properties (conjectures) :
- Monotonous
- $f(\pi)=(f(0)+f(2\pi))/2$
- $f(\pi-x)+f(\pi+x)=2*f(\pi)$, so $f(\pi)$ is a symmetric point of the curve
- First derivative is $0$ in $0$ and $2\pi$
The last point can be proven easily by deriving the equation, which gives $f'(x)=\frac{1}{4}\left(f'(\frac{x}{2})(1+\cos(\frac{x}{2}))+f'(\frac{x}{2}+\pi)(1-\cos(\frac{x}{2}))+\sin(\frac{x}{2})(f(\frac{x}{2}+\pi)-f(\frac{x}{2}+)\right)$ Evaluating at $0$ and $2\pi$ yields $0$.
Evaluating the original equation at $0$ or $2\pi$ yields consistent results, at $\pi$ gives $f(\pi)=(f(\pi/2)+f(3\pi/2))/2$, which is consistent with the second conjecture at this particular point.
So far I haven't managed to prove the other conjectures on the solution. Note that all these properties are shared by $Cos(x/2)+C$, but this is not a solution, so there are other things needed to characterize a solution. Here is a plot of an approximate numerical solution.
Any ideas how to solve this? Thanks!