I have been using factorials (a.k.a the mark "!") to calculate the number of arrangements of objects, and I'm wondering if the function f(x)=x! have a value when they are not positive integers but fractions, negative numbers, or even irrational numbers

such as the famous Euler's constant "e" or "π", and if there is an equation to calculate them out, it seems that it should be

connected to trigonometry and natural logarithm while checking the graph of it (at least for me)

Guest May 6, 2017

Do you know the gamma function? 
The informal definition of gamma function is \(\Gamma (n)=(n-1)!\)

The gamma function can be defined with the integral: \(\Gamma(z) =\displaystyle \int^{\infty}_{0}x^{z-1}e^{-x}\mathtt{dx}\) and this is the formal definition.

Gamma function have a property: \(\Gamma(x+1)=x\Gamma (x)\)

So "pi factorial" can be calculated using:

\(\pi!\\ =\Gamma(\pi + 1)\\ =\pi \Gamma(\pi)\\ =\pi\left(\displaystyle\int^{\infty}_0x^{\pi-1}e^{-x}\mathtt{dx}\right)\)

MaxWong  May 6, 2017

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