Exercise 5.46 Show that the gamma function (w) satisfies (w) = (w 1)(w 1) for w > 1, and deduce that (n) =

Exercise 5.46 Show that the gamma function Ŵ(w) satisfies Ŵ(w) = (w − 1)Ŵ(w − 1) for w > 1, and deduce that Ŵ(n) = (n − 1)! for n = 1, 2, 3, . . . .

Exercise 5.47 Let I =

Z

∞

−∞

e−x2 dx.

By changing variables to polar coordinates, show that I 2 =

ZZ R2 e−x2−y2 dx dy =

Z 2π

θ=0 Z

∞

r=0 e−r2 r dr dθ, and deduce that I = √π.