Continuous Poisson distribution: $\int_0^\infty \frac{t^n}{n!}dn$

Question

I was thinking poission distribution, actually i like it. Then i thought there is no reason for some events to be integers. We can define occurences as half finished homeworks for example, or 3.7 apples etc.

So when i give wolfram an example, it actually calculates. Mean occurence of events is 3.2, what is the probability that i see this event to be between 1 and 4 ? It is about 0.6076

My question is that although wolfram calculates, i dont know and i couldnt find this integral (all density). Can it be calculated ?

$$\int_{0}^{+\infty}\frac{t^{n}}{n!}\,dn$$

Answer 1

Since $n!=\Gamma(n+1)$, you are then asking about a closed form of $$ \int_0^\infty \frac{t^x}{\Gamma(x+1)}dx, \quad t>0. \tag1 $$ There is no known closed form of $(1)$, but this integral has been studied by Ramanujan who proved that

$$ \int_0^{\infty} \frac{t^x}{\Gamma(1+x)} \, dx = e^t - \int_{-\infty}^{\infty} \frac{e^{\large -te^x}}{x^2+\pi^2} \, dx, \quad t>0. \tag2 $$

Some useful related links are link 1, link 2, link 3.