solve $x+\sin(x)=k$ for $x$

Question

This question has been proposed to me and thus far it has baffled me:

$$ x + \sin(x) = k$$ solve for x.

Another way of looking at it is find $f^{-1}(x)$ given that $f(x)=x + \sin(x)$. Wolfram alpha doesn't even seem to know how to solve it, when you plug in a value for k, it tell you approximately the number but does not give an "exact" form. Any help?

You won't be able to solve this equation explicitly for $x$ because $\sin$ is a transcendental function. — , Apr 28 '14 at 01:22
@ASKASK As far as your problem is concerned, it means that you won't be able to solve the equation algebraically. You will have to use numerical methods, such as Newton's method. — , Apr 28 '14 at 01:27
A similar question was asked about this exact same function. — Jeff, Apr 28 '14 at 01:44

score 0 · Answer 1 · answered Apr 28 '14 at 02:19

I believe the best you can do is either solve numerically for $x$ or analytically by Lagrange inversion to get the taylor expansion

$$\sum_{n=1}^\infty \frac{k^n}{n!} \lim_{\theta\rightarrow 0^+}\left(\frac{d^{n-1}}{d\theta^{n-1}} \left( {\left(\frac{\theta}{\theta + \sin{\theta}} \right)}^n\right) \right)$$.

Also, $x + \sin(x) \ k$ is also called Kepler's equation, which is how I found this information on Wikipedia.

solve $x+\sin(x)=k$ for $x$

1 Answers1