Mathematical Methods for Physics and Engineering - Matematica.NET

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONSΓ( n)642−4−3−2−11 23 4n−2−4−6Figure 18.9The gamma function Γ(n).Moreover, it may be shown for non-integral n that the gamma function satisfiesthe important identityΓ(n)Γ(1 − n) =πsin nπ . (18.158)This is proved for a restricted range of n in the next section, once the betafunction has been introduced.It can also be shown that the gamma function is given byΓ(n +1)= √ (2πn n n e −n 1+ 112n + 1288n 2 − 139 )51 840n 3 + ... = n!,(18.159)which is known as Stirling’s asymptotic series. For large n the first term dominates,and son! ≈ √ 2πn n n e −n ; (18.160)this is known as Stirling’s approximation. This approximation is particularly usefulin statistical thermodynamics, when arrangements of a large number of particlesaretobeconsidered.◮Prove Stirling’s approximation n! ≈ √ 2πn n n e −n for large n.From (18.153), the extended definition of the factorial function (which is valid for n>−1)is given byn! =∫ ∞0x n e −x dx =637∫ ∞0e n ln x−x dx. (18.161)