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Assuming I have some relatively simple equation: $$ \eta(\alpha) = \frac{\Gamma\left(\frac{3}{\alpha}\right) \Gamma\left(8-\frac{3}{\alpha}\right)}{\Gamma\left(8\right)}, $$ is there any simple way to rearrange this and to find what $\alpha$ is, given a known value of $\eta$?

Please detail the steps for me, if this is possible, as I have several similar functions I need to do this with.

Alternatively, if there is a known numerical method to do this, rather than just performing a messy fit, that is also viable. I haven't found a function in MATLAB or Python that could do this, so I assume I'm missing something conceptually here.

Cheers.

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    $\begingroup$ You want the inverse of $\Gamma(8)\eta=-π(x-7)\dots(x-1)\csc(\pi x),x=\frac3\alpha$. This uses the reflection formula $\endgroup$ Commented Jun 5, 2023 at 12:15
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    $\begingroup$ Are $alpha$ and $nu$ real or complex? Are you interested in solutions within defined bounds of either? $\endgroup$ Commented Jun 5, 2023 at 13:05
  • $\begingroup$ @TymaGaidash is that first $\pi$ the Prime-counting function? $\endgroup$ Commented Jun 5, 2023 at 13:08
  • $\begingroup$ @horchler $\eta$ and $\alpha$ are both real. $\alpha$ should be in the limits of $2 \leq \alpha \leq 4$, and $\eta$ should be inside $0.0037 \leq \eta \leq 0.0081$, but I wish to prove what $\alpha$ is given the limits of $\eta$. $\endgroup$ Commented Jun 5, 2023 at 13:08
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    $\begingroup$ @FairyLiquid Sorry, that is $\pi\cdot (x-7)$ with multiplication $\endgroup$ Commented Jun 5, 2023 at 13:09

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With Matlab you can use fzero to efficiently invert this function numerically to obtain $\alpha$ as a function of $\eta$, provided that you specify bounds where it is monotonic:

f = @(alp, eta)gamma(3./alp).*gamma(8-3./alp)/gamma(8)-eta; eta = 0.0081; alp = fzero(@(alp)f(alp, eta), [0.75 4])

This returns $0.8992...$.

Simple plotting of $\eta(\alpha)$ over $2 \le \alpha \le 4$ in Matlab will show you that $\eta$ ranges from $0.0506...$ to $0.2809...$ for that bound.

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    $\begingroup$ As an alternative, one can do the same using the routines in scipy.optimize. $\endgroup$ Commented Jun 5, 2023 at 15:08

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