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Sorry if this is not a good question, but I normally don't venture to the math side of things, at least not that far where I can't stand anymore, so please forgive me if this question is not well formulated.

I was wondering if it is possible to find the inverse of the function $g(t)$:

$$F(t) = m_0\cdot asinh(tm_1 + a_0) + (tm_2 + a_1)(\sqrt{at^2 + bt + c})$$ $$g(t) = s = F(t) - F(0)$$

So that I have the following:

$$g^{-1}(s) = t$$

As far as I understand it there is no general way to create the inverse, but I hoped that this specific function might be an exception. I tried to use WolframAlpha to solve it but it just aborts the query.

All given variables will be set with real numbers. $t$ is expected to be in the range of $0$ to $1$, $s$ should be a positive real number, $asinh(x) = ln(x + \sqrt{x^2 + 1})$.

This function is the simplified indefinite integral of $f(t) = \sqrt{at^2+bt+c}$, where I grouped and replaced all static parts which are not including $t$, to implement it easier. I have no clue if this information helps in any way.

Please add comments if any information is missing. I will try to understand you as well as possible and try to answer all questions but neither is English my mother tongue nor do I have an extensive understanding in mathematics so there might be some information lost in translation.

Feirell
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1 Answers1

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Unfortunately, no. It does not look like this can be inverted. Rewriting the $asinh$ as a $log$ shows that we would have some term like $x^n log^m(x)$ which is not invertible.

Of course there is the inverse function theorem, which says that we can numerically find an inverse function so long as the derivative of your function, $f$ is finite and nonzero at the point of interest, which sounds like it may help you.

Note: as your function originally came from an antiderivative; it may be useful to think of the cases where you have a perfect square under the radical, and your integral simplifies dramatically.

Jacob A
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  • Thank you very much for your answer! That is a shame to hear but I somewhat expected it. You wrote that there is a numerical way, sadly I have a hard time grasping how to apply this to my specific problem, could you please provide an example or any other kind of pointer which could help me out in applying your suggestion to my problem? – Feirell Apr 04 '21 at 08:24
  • Just as an example, check out the package pynverse in python. Like I said you have to consider the inverse function. So begin by plotting the function as is to ensure we can invert it as expected on the interval(s) of your choice. Thank you for answer and bounty! – Jacob A Apr 04 '21 at 22:46