# Is $\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$ a rational number?

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 $$3$$:

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 $$\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$$

• $\frac{1}{\Gamma(x)}$ 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 0:24
• 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 4:43
• @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 5:10
• @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 5:05
• @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 5:30

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.