4 Jul
2015
4 Jul
'15
3:22 p.m.
There's a clever proof of this that's hard to forget once you see it: Let I = Integral_{-oo,oo} exp(-x^2) dx. Then I^2 = [Integral_{-oo,oo} exp(-x^2) dx] [Integral_{-oo,oo} exp(-y^2) dy] = Integral_{the xy-plane} exp(-(x^2 + y^2)) dx dy = Integral_{0,2pi} [Integral_{0,oo} exp(-r^2) r dr] dt = 2pi Integral_{0,oo} exp(-r^2) r dr = 2pi (-1/2) Integral_{0,oo} exp(-r^2) -2r dr oo = -pi exp(-r^2)] 0 = pi So I = sqrt(pi). —Dan
[the formula for the integral of exp(-x^2) as x goes from minus infinity to infinity]