Stirling's Formula: Understanding the Asymptotic Approximation

What is Stirling's Formula?

Stirling's formula is an asymptotic approximation for the factorial function. It estimates the value of n! for large values of n. This formula was first derived by James Stirling in the 18th century.

How does Stirling's Formula work?

Stirling's formula is given by:

The formula uses the Sterling approximation for the natural logarithm of the gamma function. The gamma function is defined as:

Using Stirling's formula, we can estimate the value of n! for large values of n. For example, if we want to find the value of 100!, we can use Stirling's formula and obtain an approximate value of 9.3326215e+157.

Limitations of Stirling's Formula

Stirling's formula is an approximation and is only valid for large values of n. For small values of n, the formula may not provide accurate results. The formula also does not work for complex numbers. Additionally, the approximation is not very good for small values of n, and there are other approximations that may provide better accuracy in these cases.

Conclusion

Stirling's formula is a powerful tool that is widely used in various fields such as statistics, physics, and engineering. It provides a fast and efficient way to estimate the value of factorials for large values of n. However, it is important to note that the formula has its limitations, and it may not provide accurate results for small values of n or for complex numbers.