3

I would like to learn more about the behavior of the factorial function or Gamma function, so I decided to compute the following integral $$ \int_1^\infty\dfrac{1}{\sqrt{\Gamma(x)}}\,\mathrm dx. $$

According to Wolfram alpha, its value is approximately $3$:

enter image description here

My question is whether the exact value is rational or not.

Edit: The Motivation of this question is to know more about transcendence degree of the field generated by $\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$

zeraoulia rafik
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    $\frac{1}{\Gamma(x)}$ has an integral representation, so you can try taking the integral of that to find $\int_0^{\infty} \frac{1}{\Gamma(x)}$. – Toby Mak Jul 19 '19 at 00:24
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    @TobyMak The question is about the integral of $1/\sqrt{\Gamma(x)}$ though. Does that have an integral representation? – Robert Israel Jul 19 '19 at 04:10
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    I don't understand why anyone upvoted this. Is the question if $\int_1^\infty\dfrac{1}{\sqrt{\Gamma(x)}}dx = 3$ ? The answer is no, try in pari gp intnum(x=1,[+oo,2],1/sqrt(gamma(x))) – reuns Jul 19 '19 at 04:43
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    @reuns That is not the question. The question is if it is rational or not. – Toby Mak Jul 19 '19 at 05:05
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    @TobyMak Sure and I'd like to know if the Riemann hypothesis is true, please tell me the answer. As everyone noted the OP doesn't even have an acceptable representation of his constant. – reuns Jul 19 '19 at 05:10
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    @reuns I don't see the issue with this question. Ignore the fluff about 3; is $\int_1^\infty \frac{1}{\sqrt{\Gamma(x)}}dx$ rational? What is wrong with that? – mathworker21 Jul 20 '19 at 05:05
  • @mathworker21 Does the fact that it is rational or irrational have any consequence ? If it doesn't have any consequence then it is undecidable. – reuns Jul 20 '19 at 05:26
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    @reuns I have no idea what you're saying. Isn't "undecidable" a rigorous word and "consequence" not. In any event, I'm sure the OP had some reason for asking. – mathworker21 Jul 20 '19 at 05:30
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    @mathworker21 ?? The OP didn't give any reason for asking. In particular he didn't propose any consequence of the (ir)rationality thus no reason to assume it is not undecidable. If you don't see everyday some consequences of the (ir)rationality of random constants (such as $\int_0^1 e^{-\Gamma(x) \sin(x)}dx$) it is because (for most of them) their irrationality is undecidable. – reuns Jul 20 '19 at 05:37
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    @reuns I'm just confused how giving a reason for asking changes whether it is decidable or how likely you view it to be decidable. – mathworker21 Jul 20 '19 at 05:39
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    OK, zeraoulia, the question has been reopened – now, the ball is in your court. What makes your question worth thinking about? What consequences would it have, if the number turned out to be rational? What consequences, if irrational? You want to learn more about the behavior of the Gamma function – what would the answer to your question tell you? Why the square root, and not the cube root, or the logarithm, or the cosine? I see two votes to close already, I expect there will be more if you can't do something to motivate your question. – Gerry Myerson Aug 06 '19 at 09:51
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    You missed your chance, zeraoulia. – Gerry Myerson Aug 06 '19 at 23:09
  • Concerning the most recent edit, if it turns out that the number is irrational, that will tell you exactly nothing about the transcendence degree of the field it generates. Anyway, the edit makes me wonder, why would you be interested in this particular field, of all the infinitely many fields suggested by my earlier comment. I'm still not convinced the question add anything useful to this website. – Gerry Myerson Aug 26 '19 at 06:09

1 Answers1

8

This number is somewhow similiar to the (more or less) well-known Fransén-Robinson Constant $F$. The latter is defined by a somewhat analogical integral

$$F:=\int_0^\infty\frac{\mathrm dx}{\Gamma(x)}=2.807~770\dots$$

Note that this one is close to $e=2.718~281...$ since the integral may be approximated by the standard infinite sum for Napier's constant.

However, even though the Fransén-Robinson Constant is listed in a bunch of overviews of mathematical constants there is not much more to say about this constant; and I suspect the same for your given one... It seems to be unclear whether there is an "easy" closed-form expression for the Fransén-Robinson constant in terms of other known constant and the question about irrationality is not even tossed in the room on Wikipedia for example (of course, this is a reasonable question for all mathematical constants).

Using Approach0 I have found four posts here on MSE related to the Fransén-Robinson constant

They may be of help while examining

$$\int_1^\infty\frac{\mathrm dx}{\sqrt{\Gamma(x)}}=2.992~866\dots$$

To be honest: I have doubts that this will lead somewhere.

mrtaurho
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