How to make a function that computes the factorial for numbers with decimals?

Question

How can I make a function that calculates the factorial (or the gamma function) of decimal numbers in JavaScript? For example, how could I calculate 2.33!?

Answer 1

This is not a trivial problem. There is not a simple closed-form formula for the gamma function. That said, there are some numerical approximations that should suit your needs.

The following answer will be using a technique called Lanczos approximation. The formula is as follows:

where g is an arbitrarily chosen constant that controls how accurate the approximation will be. For larger g, the approximation will be more accurate. A_g(z) is defined thus:

The hardest part is finding A_g(z), since p_n is also defined with a complicated formula dependent on g.

I can't take too much credit for the following code, since I am just writing a port of the Python program on the wikipedia page.

function gamma(n) {  // accurate to about 15 decimal places
  //some magic constants 
  var g = 7, // g represents the precision desired, p is the values of p[i] to plug into Lanczos' formula
      p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7];
  if(n < 0.5) {
    return Math.PI / Math.sin(n * Math.PI) / gamma(1 - n);
  }
  else {
    n--;
    var x = p[0];
    for(var i = 1; i < g + 2; i++) {
      x += p[i] / (n + i);
    }
    var t = n + g + 0.5;
    return Math.sqrt(2 * Math.PI) * Math.pow(t, (n + 0.5)) * Math.exp(-t) * x;
  }
}

and of course, by definition of the gamma function:

function factorial(n) {
  return gamma(n + 1);
}

You can see this in action on jsFiddle.

Answer 2

I might have found an existing solution... It's an implementation of Lanczos method, I found it at the swedish wikipedia (http://sv.wikipedia.org/wiki/Gammafunktionen). It was written in python and says to be correct up to 15 decimals. I ported it to js, cross checked some random values against (http://www.efunda.com/math/gamma/findgamma.cfm).

http://jsfiddle.net/Fzy9C/

var g = 7;
var C = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7];

function gamma(z) {

    if (z < 0.5) return Math.PI / (Math.sin(Math.PI * z) * gamma(1 - z));
    else {
        z -= 1;

        var x = C[0];
        for (var i = 1; i < g + 2; i++)
        x += C[i] / (z + i);

        var t = z + g + 0.5;
        return Math.sqrt(2 * Math.PI) * Math.pow(t, (z + 0.5)) * Math.exp(-t) * x;
    }
}

(and ofcourse it does not support imaginary numbers, since js does not)

Answer 3

Just to complete @apelsinapa answer to correct the calculation for an integer (we didn't get an integer solution when inputing an integer number).

@apelsinapa's great solution:

var g = 7;
var C = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7];

function gamma(z) {
    if (z < 0.5) return Math.PI / (Math.sin(Math.PI * z) * gamma(1 - z));
    else {
        z -= 1;

        var x = C[0];
        for (var i = 1; i < g + 2; i++)
        x += C[i] / (z + i);

        var t = z + g + 0.5;
        return Math.sqrt(2 * Math.PI) * Math.pow(t, (z + 0.5)) * Math.exp(-t) * x;
    }
}

And to get a correct answer for integer:

 function factorialOfNumber(number) {
    if (number % 1 != 0 || number<0){
        return gamma(number + 1);
    }
    else {
        if(number == 0) {
           return 1;
        }
        for(var i = number; --i; ) {
           number *= i;
        }
        return number;
    }
}