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What is the inverse of

$$f(x)=\sin(x)+x.$$

I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.

What about

$$f(x)=\sin(a \cdot x)+x$$

where $a$ is a known real constant.

Thank you for taking the time to read this question! Sorry if this has been asked before...

IV_
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Mircea
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    It's unlikely that the inverse of $f$ can be expressed using only elementary functions. Why do you need its inverse? – Git Gud Jan 26 '14 at 19:00
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    I don't think there is a nice closed form for the inverse. Depending on what you need it for, you could for instance derive Taylor expansions of the inverse at a certain point to see how it behaves locally. – TMM Jan 26 '14 at 19:01
  • @TMM I actually got to this trying to determinate this function... http://25.media.tumblr.com/tumblr_lyy8t4ihlt1qlc2l8o1_500.gif – Mircea Jan 26 '14 at 19:07
  • Maybe "determinate" that using parametric equations... – GEdgar Jan 26 '14 at 19:08
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    For what it's worth, it is called a cycloid. – Git Gud Jan 26 '14 at 19:09
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    @GEdgar determine*, sorry... – Mircea Jan 26 '14 at 19:16

4 Answers4

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Kepler's equation ... its solution is known not to be an elementary function.

GEdgar
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Why not give another “closed form” since @GEdgar’s solution for the Kepler equation technically has $E(a,M)$ as the solution in the Wikipedia link. Here is a solution to the inverse of $y=\sin(x)+x$ using mathematica function using Inverse Beta Regularized which is a standard function introduced in $1996$. The answer is from:

Closed form of $x$ for $x=\cos(x)$: Intuition for why the Dottie number is an inverse sine of the median of a Beta distribution.

where

$$M=E-e\sin(E)\iff x=y-a\sin(y)\implies y=\text E(a,x)$$

and

$$\text E(-1,x)=2\sin^{-1}\sqrt{\text I^{-1}_{\frac x\pi}\left(\frac12,\frac32\right)}=\text{hav}^{-1}\left(\text I^{-1}_{\frac x\pi}\left(\frac12,\frac32\right)\right)$$

Here is a numerical tester for the inverse function

Here is a plot of:

$$\boxed{x=\sin(y)+y\implies y=\text E(-1,x) = \text{hav}^{-1}\left(\text I^{-1}_{\frac x\pi}\left(\frac12,\frac32\right)\right),0\le x\le \pi}: $$

enter image description here

The code is inversehaversine(inversebetaregularized(x/pi,1/2,3/2))

where appears the Inverse Haversine, a transformed inverse sine function.

The differential equation for the inverse of $y=\sin(x)+x$ is:

$$y’\text{vercos}(y))=1$$

where vercos is the versed cosine, a transformed cosine function. The inverse function’s domain is extended by using a series expansion of $\text I^{-1}_z(a,b)$. Please correct me and give me feedback!

Unfortunately, it seems that no more solutions of the inverse of $x-a\sin(x)=y$ can be solved for using this method. The only exception is that the inverse of $x-\sin(x)$ can be given in closed form by switching the $\frac32$ and $\frac12$ in the boxed answer.

Тyma Gaidash
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  • Look https://math.stackexchange.com/questions/4467187/anti-dual-numbers-and-what-are-their-properties – Anixx Jun 07 '22 at 02:35
3

Using the fixed-point technique seems to be the simplest solution I found:

$y = x - \sin(y)$

$y = x - \sin(x - \sin(y))$

$y = x - \sin(x - \sin(x - \sin(y)))$

…

$y = x - \sin(x - \sin(x - \sin(x - \sin(x - ... \sin(y) ...))))$

You can assign any value to $y$ on the right side since the effect is eventually eliminated after numerous iterations of the sine operation.

Here are the plotted functions, where the $z$ slider represents the $y$ value on the right side of the equation:

sin(x) + x inverse

As observed, the source and inverse functions exhibit symmetry with respect to the line $y = x$.

Alireza Roshanzamir
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1

That's a kind of Kepler's equation. Already Liouville proved that the inverse isn't an elementary function:

$$x+\sin(ax)=f(x)$$ $$x-(-1)\sin(ax)=f(x)$$ $x\to\frac{t}{a}$: $$t-(-a)\sin(t)=af(x)$$

see How to solve Kepler's equation $M=E-\varepsilon \sin E$ for $E$?

1) Lambert W, Generalized Lambert W

$$\sin(ax)+x=y$$

$$-\frac{1}{2}i\left(e^{aix}-e^{-aix}\right)+x=+y$$

$$-\frac{1}{2}i\left(e^{aix}\right)^2+\frac{1}{2}i+e^{aix}x-e^{aix}y=0$$

$$e^{aix}=-ix\pm\sqrt{-x^2+2yx-y^2+1}+iy$$

We see, the inverse cannot be represented in closed form in terms of Lambert W either, and not by generalized Lambert W of Mezö et. al. But possibly by other generalized Lambert W - see the references below.

2) "Leal-functions"

$$\sin(x)+x=y$$ because $\sin(it)=i\sinh(t)$: $x\to it$: $$\sin(it)+it=y$$ $$i(\sinh(t)+t)=y$$ $$\sinh(t)+t=-iy$$ $$t=\text{Lsin}_2(-iy)$$ $$x=i\ \text{Lsin}_2(-iy)$$

[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553

[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)

[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018

[Stoutemyer 2022] Stoutemyer, D. R.: Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses. 2022

[Vazquez-Leal et al. 2020] Vazquez-Leal, H.; Sandoval-Hernandez, M. A.; Filobello-Ninoa, U.: The novel family of transcendental Leal-functions with applications to science and engineering. Heliyon 6 (2020) (11) e05418

IV_
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