Stirling Formula: Definition and Derivation

Stirling’s formula, also known as Stirling's approximation formula, is an approximation of relatively large numbers. Stirling formula is generally used to determine the approximate value of a factorial function (n!). There has been evidence of the Stirling formula being used for Gamma functions as well. Apart from that, it is also apparently used in applied mathematics for determining factorials of large numbers. In simple words, 

Also Read: Application of Integrals

Key Takeaways: Stirling Formula, Stirling approximations, Integers, Factorials, Formula

What is Stirling’s Formula?

[Click Here for Sample Questions]

Before understanding what Stirling's Formula is, we need to understand the meaning of a factorial. A factorial is a continuous chain of multiplication of a given positive integer with the numbers coming below it. For example, the quantity of three factorials can be determined by the following way: (3!) = 3 X 2 X 1 = 6. The symbol that helps identify factorials is “n!”, wherein ‘n’ is the known integer.

As much as there is a factorial of small numbers, there is more of larger numbers too. The issue with larger numbers is their factorial can give an extremely lengthy answer, which is what makes it unlikely to be determined. As we reach 11!, the numbers hike and consequently give results in millions, making it difficult for an individual to determine its value by mere multiplication. 

However, the issue of determining factorials of large numbers was minimized when James Stirling, a Scottish Mathematician, first introduced his now popular approximate formula. Stirling’s approximation formula gave a roughly accurate idea of the size of n!, with an error margin close to minimal. Stirling’s formula is now widely used by many to establish the factorial value of a positive integer without having to find the product by continuous multiplication.

Also Check: Permutations and Combinations

Determine Stirling’s Approximation Formula

[Click Here for Sample Questions]

Stirling’s formula or Stirling approximation is usually used to determine the approximate value of a factorial of a positive integer. Stirling’s formula came into effect after James Stirling found out that it took larger effort to determine the factorial value of comparatively bigger numbers, like for instance, “74!”.

Stirling's approximation formula helps establish the factorial of a number extremely close to the actual value. Usually, Stirling’s formula gives an error of roughly 2%, or even less oftentimes, based on the number given to determine.

Stirling’s approximation formula can be determined by the following equation: 

Stirling’s formula = n! ≈ √2πn ( n/e)ⁿ (wherein ~ denotes that the ratio of two quantities moves to 1 as n →∞) or ln n! ≈ n ln n-n (wherein ln is the natural logarithm of the equation)

Different cases use different variations of Stirling’s formula. One of the derivations of Stirling’s approximation formula uses probability distribution function which in turn establishes normal distribution for larger numbers.

Read More: Probability

Things to Remember

1. A factorial is the product of a positive integer, multiplied with the numbers coming after it. For instance, the factorial of the number 4! is 4 * 3 * 2 * 1, the product of which is typically equal to 24.
2. The symbol that helps identify factorials is “n!”, wherein ‘n’ is the known integer.
3. James Stirling, a Scottish Mathematician, first introduced the famous approximate formula after he found that determining the average value of a factorial, especially if it is a large number, becomes tedious. 
4. Stirling’s approximation formula gave a roughly accurate idea of the size of n!, with an error margin close to 2%, or even minimal. 
5. Stirling’s approximation formula can be determined by the following equation: Stirling’s formula = n! ≈ √2πn ( n/e)ⁿ or ln n! ≈ n ln n-n (wherein ln is the natural logarithm of the equation)