How to calculate the inverse factorial of a real number?

Question

Is there a way to calculate the inverse factorials of real numbers?

For example - 1.5 ! = 1.32934039 1.5 ! = 1.32934039

Is there a way to get 1.5 back if I have a value of 1.32934039 ?

I'm trying to

http://www.wolframalpha.com/input/?i=Gamma^(-1)[1.32934039]

but this is a mistake.

Answer 1

Using wolframalpha.com you can ask

 Solve[Gamma[x+1]==1.32934039,x] 

As mentioned in the comments, Gamma does not have a unique converse. True, even when you decide to use a regular factorial, for example

 Solve[Gamma[x+1]==6,x] 

gives several answers, of which 3.

Instead of using Gamma [] in WolframAlpha, you can also use Factorial []:

 Solve[Factorial[x]==6,x] Solve[Factorial[x]==1.32934039,x] 

Answer 2

David Cantrell gives a good approximation of Γ ^-1 (n) on this page :

<Preview> k = positive zero of the dimamic function, approximately 1.461632 c = Sqrt (2 * pi) / e - Γ (k), approximately 0.0.036534 L (x) = ln ((x + c) / Sqrt (2 * pi)) W (x) = Lambert Function W Approximation (x) = L (x) / W (L (x) / e) + 1/2

Answer 3

For integers you can:

 i = 2 n = someNum while (n != 1): n /= i i += 1 return (i==1 ? i : None) 

The factorial for real numbers does not have the opposite. You say that "every function should have the opposite." This is not true. Consider the constant function f(x)=0 . What is f^-1(42) ? For a function that must be inverse, it must be both injection and surjection .