Stirlingképlet

The Stirling formula, also known as Stirling's approximation, is a mathematical formula used to approximate the factorial function for large numbers. It is named after the Scottish mathematician James Stirling, who published it in 1730. The formula is given by:

n! ~ sqrt(2 * pi * n) * (n/e)^n

where n is a large number, pi is the mathematical constant approximately equal to 3.14159, and e is the base of the natural logarithm, approximately equal to 2.71828.

The Stirling formula is widely used in various fields of science and engineering, including probability theory, statistical mechanics, and information theory. It is particularly useful when dealing with very large numbers, where calculating the exact factorial becomes computationally infeasible. The approximation becomes more accurate as n increases.

The derivation of Stirling's formula often involves the use of integral calculus and the gamma function, which is a generalization of the factorial function to complex numbers. The formula can be expressed in various forms, including logarithmic forms, which are useful for numerical calculations.

While Stirling's formula provides an approximation, its accuracy is remarkably high for large values of n. The relative error of the approximation decreases as n increases, making it a powerful tool for simplifying complex calculations and analyzing the behavior of systems involving large factorials.