find an inverse function of complicated one

Question

Let $f:\mathbb{R}\rightarrow \mathbb{R}$: $$f(x) = \sin (\sin (x)) +2x$$

How to calculate the inverse of this function?
So far i searched a lot in the internet but i didn't find any easy algorithm to this.
What i found is just for easy function (like $f(x)=x^2$) , but no for complicated one.

Can someone show me the steps? are they any rules i need to know? thanks a lot in advance!

Answer 1

Two general methods exist, but often it is very hard to employ them with some success:

• integral representations as Burniston-Siewert-like representations; see: http://www4.ncsu.edu/~ces/publist.html
• series as Lagrange series: http://en.wikipedia.org/wiki/Lagrange_inversion_theorem

It is useful to remark that "simpler" equations as Kepler equation ($M=E+e\sin(E)$) needed the introduction of special functions (i.e. the Bessel functions) to be solved by means of series expansion.

For example in your case, the Lagrange inversion would give the following formal series solution near 0:

$$x(u)= \frac{u}{2} + \sum_{n=1} \frac{(-1)^{n}}{2(n!)}\left(\frac{d}{du}\right)^{n-1}\sin\left(\sin\left(\frac{u}{2}\right)\right)^n $$