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. ${\displaystyle \int _{1}^{n+{\frac {1}{2}}}{\ln(x)}\;\mathrm {d} x}$

See also

• Approximate calculation of the gamma function

literature

• Claudi Alsina, Roger B. Nelsen: Enchanting Evidence: A Journey Through the Elegance of Mathematics . Springer Spectrum , Berlin (among others) 2013, ISBN 978-3-642-34792-4 , p. 269, 306-307 .
• Francis J. Murray : Formulas for Factorial N . In: Mathematics of Computation . tape 39 , 1982, pp. 655–661 ( online copy [PDF]).
• William Burnside : A rapidly convergent series for Log N! In: The Messenger of Mathematics . tape 46 , 1917, pp. 157-159 .

Individual evidence

1. ^ Claudi Alsina, Roger B. Nelsen: Enchanting Evidence: A Journey Through the Elegance of Mathematics. 2013, pp. 269, 306-307
2. ↑ ^{a b} Alsina / Nelsen, op.cit., P. 269
3. ^ Francis J. Murray: Formulas for Factorial N. Math. Comp. 39: 655-661 (1982)
4. ↑ Alsina / Nelsen, op.cit., P. 306