# Burnside formula

The formula of Burnside is a formula of mathematical sub-region of the analysis , which the English mathematician William Burnside back. It is closely related to Stirling's formula and, like this, gives an approximation of the factorial function .

## Representation of the formula

Burnside's formula can be given as follows:

${\ displaystyle n! \ sim {\ sqrt {2 \ pi}} \; \ left (\ n + {\ frac {1} {2}} \ right) ^ {\ left (\ n + {\ frac {1} { 2}} \ right)} {\ mathrm {e}} ^ {- \ left (\ n + {\ frac {1} {2}} \ right)} = {\ sqrt {2 \ pi}} \; \ left ({\ frac {n + {\ frac {1} {2}}} {\ mathrm {e}}} \ right) ^ {\ left (\ n + {\ frac {1} {2}} \ right)}}$

## Goodness of approach

Claudi Alsina and Roger B. Nelsen refer in their monograph Bewitching Evidence (Springer, 2013) to the fact that Burnside's formula is “approximately twice as accurate as Stirling's formula” and that its derivation “can be derived from approximations for the integral “Wins. ${\ displaystyle \ int _ {1} ^ {n + {\ frac {1} {2}}} {\ ln (x)} \; \ mathrm {d} x}$