Closed-form formula for inverse of this function

Question

I need to find the inverse of the expected value of the continuous Bernoulli distribution. Here is the formula:

$$ \frac{x}{2x-1} + \frac{1}{2\tanh^{-1}(1-2x)} $$

Is there a simple closed-form formula for inverse of this function?

Answer 1

Your formula is the function term of an elementary function.
This function doesn't have partial inverses that are elementary functions.

$$\text{arctanh}(x)=\frac{1}{2}\,\ln(x+1)-\frac{1}{2}\,\ln(1-x)$$

$$\frac{x}{2\,x-1}-\frac{1}{2\,\text{arctanh}(2\,x-1)}$$

$$\frac{x}{2\,x-1}-\frac{1}{2\,(\frac{1}{2}\,\ln(2\,x)-\frac{1}{2}\,\ln(2-2\,x))}$$

$${\frac{-x\ln(x)+x\ln(1-x)+2\,x-1}{(2\,x-1)(-\ln(x)+\ln(1-x))}}$$

This is the function term of an elementary function of $x$. It's the function term of an algebraic function over the algebraic numbers of $x$, $\ln(x)$ and $\ln(1-x)$. If Schanuel's conjecture is true, the transcendence degree of $\{x,\ln(x),\ln(1-x)\}$ over the algebraic numbers is 3. We cannot simplify the algebraic term of the three indeterminates to one with less than 3 indeterminates therefore. With help of the theorem in [Ritt 1925], that is proved also in [Risch 1979], we can conclude that the mentioned elementary function doesn't have partial inverses that are elementary functions.
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[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759

[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90