I am looking for X if I know y and:
Gamma(X)=y
Any help is highly appreciated.
2026-03-31 12:44:50.1774961090
What is X if I know y and Gamma(X)=y?
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Trivial for starting $$\Gamma(x)=y\implies x=\Gamma^{(-1)}(y)$$ Now, have a look here and admire the simplicity and beauty of David W. Cantrell's solution. It is really impressive.
Defining
$$L(y)= \log \left(\frac{c+y}{\sqrt{2 \pi }}\right)$$ $$\Gamma^{(-1)}(y)=\frac{L(y)}{W\left(\frac{L(y)}{e}\right)}+\frac{1}{2}$$ (using $c=0.036534$) or more rigorously $$c=\frac {\sqrt{2\pi}} e- k$$ $k$ being the positive zero of the digamma function $(\approx 1.461632)$.