faktoriály

Faktoriály, in mathematical usage, refers to the factorial function and related concepts. For a nonnegative integer n, the factorial n! is defined as the product of all positive integers from 1 to n: n! = 1 × 2 × ... × n, with 0! = 1. The factorial grows rapidly with n and models the number of permutations of n distinct objects.

Factorials have important combinatorial and algebraic roles. The factorial counts the number of ways to arrange n distinct items, and binomial coefficients are expressed as n!/(k!(n−k)!). This leads to formulas such as the number of k-element subsets of an n-element set, written as C(n, k) = n!/(k!(n−k)!). Factorials also appear in series expansions, including those for exponentials and trigonometric functions.

Beyond nonnegative integers, factorials extend to real and complex numbers via the gamma function: n! = Γ(n+1). The gamma function satisfies Γ(z+1) = zΓ(z) and is defined for complex numbers with positive real part, with poles at negative integers. This extension allows a continuous analogue of factorials and connects to many areas of analysis and number theory.

Growth and approximation are central in analysis. Stirling’s formula provides an asymptotic approximation: n! ~ sqrt(2πn) (n/e)^n for large n. Logarithmic forms, such as log n! ≈ n log n − n, are used to manage large values in computations and proofs.

Variants include the double factorial n!! and rising factorial (x)_n (Pochhammer symbol). In computation, factorial values become impractically large for moderate n, often requiring arbitrary-precision arithmetic or logarithmic representations.