Fakultetsfunktionen

Fakultetsfunktionen, often denoted n!, assigns to each nonnegative integer n the product 1×2×...×n, with 0! defined as 1. It counts the number of ways to arrange n distinct objects in a line (permutations) and appears in formulas such as binomial coefficients and probability calculations.

Analytically, the factorial function extends beyond integers through the gamma function Γ(z). For integers, n! = Γ(n+1). The gamma function is defined by the improper integral Γ(z) = ∫_0^∞ t^{z-1} e^{-t} dt for Re(z) > 0 and satisfies the recurrence relation Γ(z+1) = z Γ(z). It has simple poles at z = 0, -1, -2, ..., and provides a smooth extension of the factorial to positive real and complex numbers via n! = Γ(n+1).

Key properties include the recurrence n! = n × (n−1)!, the initial condition 0! = 1, and the rapid growth of factorials, which outpace any exponential function. A standard asymptotic approximation is Stirling’s formula: n! ~ sqrt(2πn) (n/e)^n as n → ∞, useful in analysis and approximations.

Combinatorially, n! is the number of permutations of n distinct objects. It features in binomial coefficients through n choose k = n! / (k!(n−k)!), and in various counting formulas and probability distributions. Related concepts include the double factorial n!!, multifactorials, and the rising factorial (n)_k, which connects to gamma via (n)_k = Γ(n+k)/Γ(n). The factorial function thus sits at the intersection of combinatorics, analysis, and applied mathematics.