Let's round this off..

1. For [$]x[$] not a multiple of [$]\pi[$] the problem has a unique solution, contradicting earlier incorrect statements.

2. The Bessel function expansion 1817 converges to the wrong value (and slow) for large [$]\varepsilon[$].

3. The only hope is to write the equation as a least squares interval search (the function is unimodal). And very fast.

4. I also have no hope for Newton Raphson et al either.

example

[$]\varepsilon = 13.5, x = 0.8[$],

[$]y = -0.064047..[$]

You can check by plugging this value into the original equation.

.

//

[$]\varepsilon = 500.0, x = 0.8[$] ??