How to find the factorial of fractional numbers using matlab code?

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I have an array s1. I want to evaluate the factorial of s1.
M=3;
m1 = 0:M;
s1 = m1./2
s1 = 1×4
0 0.5000 1.0000 1.5000
factorial(s1)
Error using factorial
N must be an array of real non-negative integers.

Accepted Answer

John D'Errico
John D'Errico on 16 Jul 2022
You CANNOT compute the factorial of a fractional number. Factorials are defined only for integers.
HOWEVER...
It is true that
factorial(N) == gammma(N+1)
for integer N. You might think of the gamma function as an extension of the factorial function onto the real line. For example:
N = 0:10;
factorial(N)
ans = 1×11
1 1 2 6 24 120 720 5040 40320 362880 3628800
gamma(N + 1)
ans = 1×11
1 1 2 6 24 120 720 5040 40320 362880 3628800
And the gamma function is defined on the real line. So...
gamma(1.5)
ans = 0.8862
is thus what you might think of when you write (0.5)!.
fplot(@gamma,[.1,5])
In fact, the gamma function follows the smiple rules that work for factorial. Thus we would see that if
factorial(N) == N*factorial(N-1)
then we might hope it would be true that
gamma(N+1) = N*gamma(N)
For example, we can test that as:
format long g
[gamma(3.25)*3.25,gamma(4.25)]
ans = 1×2
8.28508514183522 8.28508514183522
So the gamma function follows a similar recursive rule like the factorial function. The only differnce is you don't have any easy way to start the recursion.
  2 Comments
Steven Lord
Steven Lord on 18 Jul 2022
This sounds like a reasonable suggestion. I've added it to the enhancement database.

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