Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSIf we let x = n + y, then(ln x =lnn +ln 1+ y )n=lnn + y n − y22n 2 + y33n 3 − ··· .Substituting this result into (18.161), we obtain∫ ∞[n! = exp n(ln n + y ···) ]−nn − y22n + − n − y dy.2Thus, when n is sufficiently large, we may approximate n! by∫ ∞n! ≈ e n ln n−n e −y2 /(2n) dy = e n ln n−n√ 2πn = √ 2πn n n e −n ,−∞which is Stirling’s approximation (18.160). ◭The beta function is defined by18.12.2 The beta functionB(m, n) =∫ 10x m−1 (1 − x) n−1 dx, (18.162)which converges for m>0, n>0, where m and n are, in general, real numbers.By letting x =1− y in (18.162) it is easy to show that B(m, n) =B(n, m). Otheruseful representations of the beta function may be obtained by suitable changesof variable. For example, putting x =(1+y) −1 in (18.162), we find that∫ ∞y n−1 dyB(m, n) =. (18.163)0 (1 + y)m+nAlternatively, if we let x =sin 2 θ in (18.162), we obtain immediatelyB(m, n) =2∫ π/20sin 2m−1 θ cos 2n−1 θdθ. (18.164)The beta function may also be written in terms of the gamma function asB(m, n) = Γ(m)Γ(n)Γ(m + n) . (18.165)◮Prove the result (18.165).Using (18.157), we haveΓ(n)Γ(m) =4=4∫ ∞0∫ ∞ ∫ ∞0 0x 2n−1 e −x2 dx∫ ∞0y 2m−1 e −y2 dyx 2n−1 y 2m−1 e −(x2 +y 2) dx dy.638