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I have a function which has the following form:

$$ f(x)=K_1 \coth (Q_1 Q_2 \sqrt{x})^2 + \frac{1}{x}\left[K_2 + K_3 \coth(Q_1 Q_3\sqrt{x})\sqrt{x}\right]$$

and I want to find $x$ for $f(x)=1$. I'm fairly convinced that this will not work without approximations (feel free to correct me!). I have tried two different approximations in the limit of small $Q_1$, a Taylor series and a Pade approximation, in which I only get a manageable equation (i.e. at maximum a few lines) for first order in $Q_1$ in case of the Taylor expansion and (0,1) order in case of the Pade approximant. These two approximations are unfortunately not very good so I am looking for other approximation methods that I can try.

What I would like to know is whether there are any other appropriate approximations that I could apply to this function?

Maybe useful information: $x$ has a physical meaning such that it should be positive and real

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Is there a specific reason for needing a function which approximates it as opposed to an iterative solver (e.g. Newton-Raphson)? – Peter Taylor Oct 15 '13 at 15:16
Yes, I would like to have an approximate solution to get a feel for the way the result scales with $K_i$ and $Q_i$. Without having to scan a whole range of these parameters. – Michiel Oct 15 '13 at 15:18
About how large are the parameters? I get a pretty good estimate for the root when $K_1 = 1/10$, $K_2 = K_3 = Q_2 = Q_3 = 1$, and $Q_1 \ll 1$ by using a Taylor expansion for small $Q_1$. – Antonio Vargas Oct 17 '13 at 18:50

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