Periodic function: y(x + Cos[x]) = Cos[x]

[ksr]

Hello,

This is something I encountered while working on a research problem.

I have a periodic function whose shape I have determined to be of the form:

y(x + Cos[x]) = Cos[x]

Here, x is the free variable and 'y' denotes the displacement as a function of x (In my problem y(x) denotes the topology of a surface).

If we plot y(x) from the above equation in Mathematica one notices a periodically varying function - looks similar to a sine curve but somewhat 'slanted'.

I am trying to find an analytical expression for y(x) from the above equation.

Here is my attempt:

Put x + Cos[x] = z;

I get
y(z) = Cos[x]
i.e., y(z) = z - x
But x = ArcCos[y(z)]

Thus we have:
y(z) = z - ArcCos[y(z)]

Is it possible to solve for y(z) from this ? It has to be periodic.

Thanks!