How was Stirling's approximation derived?

Thread starter Paradox
Start date Apr 23, 2003
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Approximation

In summary, Stirling's approximation was derived by mathematician James Stirling in the 18th century. He observed that as the number n gets larger, the factorial of n (n!) approaches the value of √(2πn)(n/e)^n. This led to the development of Stirling's formula, which is used to approximate large factorials and has applications in various fields, including statistics and physics. Stirling's approximation has since been refined and expanded upon, but it remains a fundamental tool in mathematical analysis and calculation.

Apr 23, 2003

Paradox

I was wondering how Stirling's approximation

x! ~ sqrt(2[pi]x)x^xe^-x

was derived. Anyone know?

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Apr 23, 2003

Big Jeff

This article at Mathworld derives Stirling's approximation step by step.

Apr 24, 2003

Paradox

Thanks for the link, Jeff. It helped a lot

1. What is Stirling's approximation?

Stirling's approximation is a mathematical formula used to approximate the value of factorials for large numbers. It is named after Scottish mathematician James Stirling.

2. How does Stirling's approximation work?

Stirling's approximation uses a series expansion to approximate the factorial of a large number. It is based on the fact that as the number gets larger, the ratio of the factorial to the square root of the number approaches a constant value.

3. What is the formula for Stirling's approximation?

The formula for Stirling's approximation is n! ≈ √(2πn) * (n/e)^n, where n is the large number whose factorial is being approximated, π is the mathematical constant pi, and e is the mathematical constant Euler's number.

4. What is Stirling's formula used for?

Stirling's approximation is used to calculate the approximate value of factorials for large numbers, which can be useful in various mathematical and scientific applications. It can also be used to estimate the values of other mathematical functions, such as the gamma function.

5. What are some limitations of Stirling's approximation?

Stirling's approximation is only accurate for large values of n. For small values, it can produce significant errors. It also does not work well for values close to zero or negative values. Additionally, it is an approximation and not an exact value, so it may not be suitable for situations that require precise calculations.