MHB Can you solve for y in sin(y) - y = x√(2)?

[wheepep]

sin(y) - y = x√(2)
solve for y

 

[skeeter]

What makes you think a closed-form solution for y can be found?

 

[HOI]

Replacing [math]sin(y)[/math] by [math]\frac{e^{iy}- e^{-iy}}{2i}[/math] you might be able to get a solution in terms of the "Lambert W function" https://en.wikipedia.org/wiki/Lambert_W_function.

 

[Theia]

Maybe there exists a closed form solution exists, maybe not... But could I ask, what is the purpose for searching such solution?

I know, in celestial mechanics, the Kepler equation is same as this - at least after a change of variables - which means, you can take a look at literature, if you find something about it. As being so, I wouldn't use my time to kick the equation, as it is one of the most researched one in the world. Unless I wanted some sort of challenge, of course.

Anyhow, your choices for the solution will most likely be an iterative method, a power series or a Fourier series or interpolation of the numerical solution over the values of x under interest.

 

[Greg]

In general, periodic functions are of interest due to their frequent occurrence in natural phenomenon. As speculation, this particular function may be of interest due the times and places it occurs.