The Wikipedia page on Stirling's formula gives two proofs. They are not completely free of shortcuts, but they provide links to details of all the missing steps.Pratibha said:hey, can someone suggest where will a get a detailed proof of stirling's formula (related with gamma function). I read it from PRINCIPLES OF MATHEMATICAL ANALYSIS Walter Rudin,but in it many shortcut steps are there, which is either beyond my knowledge, or i m not getting it.
Stirling's Formula, also known as Stirling's approximation, is an approximation for the factorial function. It is commonly used to estimate large factorials, which can be difficult to calculate directly. The formula was first derived by Scottish mathematician James Stirling in the 18th century.
Stirling's Formula uses a continuous function to approximate the factorial function. It involves taking the natural logarithm of the factorial and then simplifying the expression. The resulting formula is much easier to compute and provides a close estimate to the actual factorial value.
Stirling's Formula is important because it allows us to estimate large factorials without having to calculate them directly. This can be useful in various mathematical and scientific applications, such as in probability and statistics, physics, and computer science.
The detailed proof of Stirling's Formula involves using the Euler-Maclaurin summation formula to approximate the factorial function. This involves breaking the factorial into smaller parts and using integration to simplify the expression. The full proof can be found in many mathematical textbooks and online resources.
While Stirling's Formula is a useful approximation, it does have its limitations. It becomes less accurate as the factorial value increases, and it is only an approximation, not an exact value. Additionally, it may not work well for negative or non-integer values. It is important to understand the limitations and use appropriate methods for calculating factorials in different situations.