# Does the Gamma Function have an Inverse?

Does the Gamma Function have an Inverse? (Is there an "arc-gamma" function?)

Where $\Gamma(x) = y... \Gamma^{-1}(y) = x\ (arc\Gamma(y)=x)$.

I've searched and found something called DiGamma Function, but when I substituted it didn't seem to be an inverse ("arc") but something else. I am not yet developed enough to understand haha...

Edit, I'd like to draw special attention to the comment below by u/G Cab, be sure to check it out, has a very useful answer to this question, but comments often go overlooked.

• You can define an inverse function on any interval on which $\Gamma$ is monotone, say $x\geq1$. – Yuval Filmus Sep 15 '14 at 3:18
• @YuvalFilmus but how do I evaluate it? – Albert Renshaw Sep 15 '14 at 3:29
• That's a current question. There are many ways. You can use binary search, for example, using some good lower and upper bounds for $\Gamma$ which can be inverted explicitly. – Yuval Filmus Sep 15 '14 at 3:31
• @YuvalFilmus Okay, so there is no like known integral or anything, it's all just iterating guesses until you get the number of decimal places you need. – Albert Renshaw Sep 15 '14 at 5:59
• – G Cab Sep 3 '18 at 23:10

## 1 Answer

Is there an arc-gamma function ?

Of course there is. Is it also expressible in terms of elementary functions ? No. $($I believe that this is what you were ultimately trying to ask$)$.

Sorry if "arc" is the wrong term.

It is! But hey, life's too short to be sorry. ;-)

I've searched and found something called DiGamma Function.

The digamma, trigamma, and polygamma functions are the derivatives of the $\Gamma$ function, not its inverse.

• Thankyou for your very kind answer :D Okay, so the answer is that there is CURRENTLY no easily-expressible inverse-gamma function? – Albert Renshaw Sep 15 '14 at 5:58
• @AlbertRenshaw: No. It can be determined $($though not by me$)$, using certain elements of abstract algebra $($which I never actually studied, but of whose existence I am aware$)$, whether a function possesses or not an inverse and/or antiderivative expressible as a finite combination of elements of a specific set of given functions. However, if the function is analytical $($and $\Gamma$ certainly is$)$, then the Lagrange inversion theorem can be employed to express that inverse as an infinite series. – Lucian Sep 15 '14 at 6:28
• @Lucian Could you tell me where I could find a wikipedia page/pdf/the name/etc about this abstract algebra proof to show if a function possesses an inverse expressible as a finite combination of elements of a specific set of given functions? – P-S.D Jun 11 '17 at 10:07
• @P-S.D: Unfortunately, I am not able to remember anymore. At a first glance, Lagrange's inversion theorem seems to come pretty close, but, in and of itself, appears rather useless. However, if one were to use it in conjunction either with his reversion theorem, or with Liouville's theorem, in the eventuality that the infinite power series given by the former might be shown to satisfy... (to be continued) – Lucian Jun 12 '17 at 4:24
• @P-S.D: ...certain implicit functional and/or differential equations “attackable” by the latter two, then all these things taken together might ultimately amount to precisely just such a theorem. – Lucian Jun 12 '17 at 4:25