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I'm sure that's a coincidence, but the Laplace transform of $1/\Gamma(x)$ at $s=1$ turns out to be pretty close to the inverse of the Golden ratio:

$$F(1)=\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx=0.61985841414477344973$$

Can we prove analytically that: $$\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx>\frac{1}{\phi}$$

The continued fraction of this number also starts very beautifully:

$$F(1)=\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\dots}}}}}}}}}}=$$

$$=[0; 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 7, 2, 1, 1, 1, 2, 6, 1, 4, 7, 1, 3, 1, 1, 6, 1,\dots]$$

The question has no practical application, but everyone here loves Golden ratio questions, right?

I'm sure there is no closed form for this integral, but if there is some useful transformation, I would like to see it as well.

If we replace every partial quotient in the CF after the first 5 2's by $2$, we obtain a great approximation to the integral:

$$F(1) \approx \frac{1}{41} \left(24+\sqrt{2}\right)$$

So, the more complicated question would be:

Can we prove analytically that: $$\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx<\frac{1}{41} \left(24+\sqrt{2}\right)$$

The difference is about $4 \cdot 10^{-7}$.

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    I think we may exploit the log-convexity of the $\Gamma$ function, but we have to be pretty careful in order to achieve such accuracy goal. – 2017-01-17
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    How about product form of gamma function? – 2017-01-18
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    @SimpleArt, I tried some partial products of the Weierstrass product for Gamma function, but even though they are integrated in closed form, the resulting expressions don't really tell me anything, except for the fact that the integral is less than $1$. – 2017-01-18
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    :-/ I don't have time to try it myself, so I'll look more later. – 2017-01-18
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    The manipulations shown by the *addendum* [here](https://math.stackexchange.com/a/2113495/44121) might be useful, too. – 2018-01-09
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    @JackD'Aurizio, thank you for the reference, the results in the linked post are pretty impressive – 2018-01-10

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