Factorials about decimals

Question

How do you get the factorial of a decimal number using a pen and paper if it is possible?

Example: Find the factorial of $0.5!$

Answer 1

You don't calculate $0.5!$ by hand. Decimal factorials do not make sense. The Gamma function might do what you want, but it's not the easiest to do by hand as it's a pretty unfriendly integral, at least in general. You might be able to calculate $\Gamma(1.5)$ (which would corespond to $0.5!$) by hand if you really wanted to: $$ \Gamma(1.5) = \int_0^\infty \sqrt x e^{-x} dx $$ and it turns out to be $\frac12\sqrt\pi$.

Answer 2

You can use Stirling's approximation, but it won't work well for small numbers like $\frac12$. For larger numbers, Stirling's approximation is quite good whether for integers or non-integers. It says: $$n! \approx \sqrt{2\pi n}\Bigl(\frac ne\Bigr)^n.$$

For example, $12.7!\approx 2.8616\times10^9$ and the Stirling formula gives around $2.843\times10^9$.

Of course, how to evaluate Stirling's formula with pencil and paper is an interesting question all by itself.

For smaller numbers of the form $n+\frac12$ your best bet is just to know that $$\frac12! = \frac12\sqrt\pi$$ and then use the rule that $(x+1)! = (x+1)x!$ so for example $\frac32! = \frac32\cdot \frac12! = \frac34\sqrt\pi$.

For numbers that are not half-integers, I have no good suggestions. There may be something involving the reciprocal factorial function (that is, $x\mapsto \frac1{x!}$) that can be calculated with some accuracy. There is a MacLaurin series that will be accurate, but it does not look easy to calculate with pen and paper.