Inverse Factorial

After I had a conversation with a friend on a very long bus ride about inverse factorial a stork came and put this page on the web. 

 My Pages:

- Forsooth

-Resume

Before we look at inverse factorial lets look at factorial:

~The most commonly used definition of factorial is only defined for non negative integers. This definition of factorial is:
n!= n*(n-1)*(n-2)*(n-3)…3*2*1

Note:    0!=1

~However as the above definition is not defined for non integers or negative values I will use:
n!= Gamma(n-1) 

where gamma is defined as:

  

Or in words: the area under the curve of  

  

 

with values of t ranging from 0 to infinity.

~For more on the gamma function go to:

Wolfram

~Looking at Inverse factorial:

~Inverse factorial is just the inverse function of factorial

~I would use Γ-1 (n) to define the inverse gamma function     however I have seen n? usedt to mean inverse factorial (this comes from Halfbakery)

~Now Lets look at what the factorial function looks like:

   ~ x {0,3}
 

   ~ x {0,10}

 

   ~And my personal favoritex: {0,100}


~And here are the graphs of Inverse Factorial:

   ~ y {0, 3}

 

   ~ y {0, 10}

 

   ~ y {0, 150}
 

 ~A note: be careful inverse factorial is not a function so there are multiple values for any one x (unless you limit the domain to x>/=1)