Inverse of $\frac{\sin(x)}{x}$

Question

How would one find the inverse of the function $y=\frac{\sin(x)}{x}$? Here are my steps: $y=\frac{\sin(x)}{x}$, $x=\frac{\sin(y)}{y}$, $xy=\sin(y)$, $\arcsin(xy)=y$, After that step, I can’t find a way to isolate $y$.

Answer 1

If you are concerned by the inverse function, you could use the usual Taylor series of $\sin(x)$ and use series reversion to get $$x=t+\frac{1}{40}t^3+\frac{107 }{67200}t^5+\frac{3197 }{24192000}t^7+\frac{49513 }{3973939200}t^9+O\left(t^{11}\right)$$ where $t=\sqrt{6(1-y)}$.

To see how good or bad it is, give $x$ a value from which you obtain $y$ and recompute $x$ from the expansion. Below are given some results using the above truncated series $$\left( \begin{array}{ccc} x_{given} & y & x_{calc} \\ 0.0 & 1.00000 & 0.00000 \\ 0.1 & 0.99833 & 0.10000 \\ 0.2 & 0.99335 & 0.20000 \\ 0.3 & 0.98507 & 0.30000 \\ 0.4 & 0.97355 & 0.40000 \\ 0.5 & 0.95885 & 0.50000 \\ 0.6 & 0.94107 & 0.60000 \\ 0.7 & 0.92031 & 0.70000 \\ 0.8 & 0.89670 & 0.80000 \\ 0.9 & 0.87036 & 0.90000 \\ 1.0 & 0.84147 & 1.00000 \\ 1.1 & 0.81019 & 1.09997 \\ 1.2 & 0.77670 & 1.19995 \\ 1.3 & 0.74120 & 1.29989 \\ 1.4 & 0.70389 & 1.39980 \\ 1.5 & 0.66500 & 1.49964 \\ 1.6 & 0.62473 & 1.59937 \\ 1.7 & 0.58333 & 1.69896 \\ 1.8 & 0.54103 & 1.79834 \\ 1.9 & 0.49805 & 1.89741 \\ 2.0 & 0.45465 & 1.99608 \\ 2.1 & 0.41105 & 2.09421 \\ 2.2 & 0.36750 & 2.19165 \\ 2.3 & 0.32422 & 2.28819 \\ 2.4 & 0.28144 & 2.38362 \\ 2.5 & 0.23939 & 2.47768 \\ 2.6 & 0.19827 & 2.57009 \\ 2.7 & 0.15829 & 2.66053 \\ 2.8 & 0.11964 & 2.74866 \\ 2.9 & 0.08250 & 2.83412 \\ 3.0 & 0.04704 & 2.91653 \end{array} \right)$$

For sure, we couls make it better using more terms.

Another possibility could be to tansform the above series as a Padé approximant to get $$x=t\,\frac {1-\frac{2927561 }{27485040}t^2+\frac{193184137 }{138524601600}t^4 } {1-\frac{3614687 }{27485040}t^2+\frac{428067253 }{138524601600}t^4 }$$

Answer 2

Elementary functions:

$$f(x)=\frac{\sin(x)}{x}$$ $$f(x)=-\frac{1}{2x}i(e^{ix}-e^{-ix})$$

We see, this function is an algebraic function in dependence of both $x$ and $e^x$. Liouville proved that such kind of functions (over a complex domain without isolated points) don't have (partial) inverses that are elementary functions: How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?

Lambert W, Generalized Lambert W:

The defining equation for the inverse $\frac{\sin(x)}{x}$ can be rearranged to a polynomial equation of both $x$ and $e^x$ which is quadratic for $e^x$. This equation is therefore not in a form to apply Lambert W or Generalized Lambert W.

"Leal-functions":

The partial inverses of the function mentioned in the question can be represented in terms of the function $\text{Lcsc}$ presented in [Vazquez-Leal et al. 2020].

$$\frac{\sin(x)}{x}=y$$ $\sin(x)=\frac{1}{\csc(x)}$: $$\frac{1}{x\csc(x)}=y$$ $$x\csc(x)=\frac{1}{y}$$ $$x=\text{Lcsc}\left(\frac{1}{y}\right)$$

We can take the "Leal functions" as closed-form functions because some of their algebraic properties and their applicability for some other kinds of equations are presented in the cited article.

[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