The Inverse of Factorials: Exploring the Gamma Function and Beyond

Factorials are a fundamental concept in mathematics, often used in combinatorics, algebra, and statistical distributions. The factorial of a non-negative integer n, denoted n!, is the product of all positive integers up to n. While basic arithmetic operations like addition, subtraction, multiplication, and division have their respective inverses, what about factorials? Can we find an inverse operation that undoes the factorial?

Understanding Factorials

The factorial of a non-negative integer n is defined as:

n! n × n-1 × n-2 × ... × 1

The Inverse of Factorial

The inverse operation for factorials is more complex than a simple arithmetic operation. Instead, it is related to the concept of the Gamma function. For non-negative integers, the Gamma function Γ(n) is defined as:

Γ(n) (n-1)! for positive integers

Thus, for non-negative integers, we can express n! in terms of the Gamma function as:

n! Γ(n 1)

Finding the Inverse

To find the inverse of the factorial, we typically use the inverse of the Gamma function, although this is not an elementary function and cannot be expressed in a simple formula. The Gamma function's inverse, often denoted as -1Γ(x), is not elementary and requires numerical methods or approximations for specific values.

Stirling's Approximation

For practical purposes, especially when dealing with large values of n, we can use Stirling's approximation:

n! ≈ √{2πn} (n/e)n

This approximation can help estimate n given n!. By solving Stirling's formula for n, we can obtain:

n ≈ ln(n!) / ln(e) ln(1/√(2πn))

Conclusion

While there is no direct formula to find n given n! for any arbitrary n, various methods can provide close approximations for practical purposes.

For instance, in the past, I derived an approximation that provides exact results for n given n! for n in the range [2, 170]. This approximation is particularly useful for computational purposes, although it may diverge for large Γ(x) values. The formula is proven stable for factorials up to 170!, which is an incredibly large number.

For readers with superior computational capacity, further investigation into when this approximation deviates significantly could be fascinating. Understanding the limitations and improvements of such approximations is crucial in fields that heavily rely on factorials, such as combinatorics and statistical analysis.