Factorials I
I have a fascination with factorials that matches my fascination with the Fibonacci and Lucas series. The factorial of a real non-negative integer is calculated on the basis that the factorial of 0-conventionally represented as 0!-equals 1, and n!=n((n-1)!).
Thus:
1! = 1
2! = 1 × 2 = 2
3! = 1 × 2 × 3 = 6
4! = 1 × 2 × 3 × 4 = 24
5! = 1 × 2 × 3 × 4 × 5 = 120
...
Some years ago I realised that non-integers should also have factorials. With the help of my computer, and using progressively refined formulae I worked out ½! to fifteen decimal places. It came to 0.886226925452758. Subsequently, acting on a tip from my son-in-law to be, I worked out that this number is half the square root of pi. Lately I've been working on the factorials of real negative non-integers (the factorials of negative integers being infinite). I've also been wondering about the factorials of complex numbers. I have a calculator that can operate with complex numbers, and the Windows calculator can work out factorials of non-integers, but I have no way of combining the two.