A cleaner, but much trickier, derivation of formula B.25 is similar to the method used in Section B.1 to evaluate the basic Gaussian integral. The trick is to consider the integral of the function e−r2, over all space, in d dimensions.
(a) First evaluate this integral in rectangular coordinates. You should obtain πd/2.
(b) Because the integral has spherical symmetry, you can also evaluate it in d-dimensional spherical coordinates. Explain why the angular integrals must give a factor of Ad(1), the area of a d-dimensional unit hypersphere. Thus, show that the integral is equal to Ad(1)⋅∫0∞rd−1e−r2dr.
(c) Evaluate the integral over r in terms of the gamma function, and thus derive equation B.25.