MA670 Russell Brown

Below are revised versions of the homework problems. Sorry about the mistakes.

1. Chapter 6, #10 continued.

Use the change of variables s = t^{2} to express

∫_{0}^{1}(1 − t^{2})^{(n−1)/2} dt  (1)

in terms of the beta function. The beta function is defined for a > 0 and b > 0 by

B(a, b) = ∫_{0}^{1}(1 − s)^{a−1}s^{b−1} ds.

Use the formula

B(a, b) = Γ(a)Γ(b)/Γ(a + b)

to express the integral (1) in terms of the Γ function.

Derive and prove a formula for v_{n}, the volume of the unit ball in R^{n}.

2. Assume that f ≥ 0 on (0, ∞). Consider the integral I defined by

I = ∫_{{x:x_{k}>0,k=1,...,n}} f(x_{1} + … + x_{n})x_{1}^{a_{1}−1} … x_{n}^{a_{n}−1} dx.

(a) Show that f(x_{1} + … + x_{n}) is measurable on R^{n}. (Hint: Imitate the proof that f(x − t) is measurable on R^{2n} in our study of convolutions.)

(b) Use a linear change of variables t = x_{1} + … x_{n}, and t_{i} = x_{i}, i = 1, …, n − 1 and then a scaling to conclude that

I = ∫_{0}^{∞} f(t) dt
× ∫_{{t∈R^{n−1}:t_{1}+...t_{n−1}<1,t_{i}>0}}(1 − t_{1} − … − t_{n−1})^{a_{n}−1}
×t_{1}^{a_{1}−1} … t_{n−1}^{a_{n−1}−1} dt_{1} … dt_{n−1}.

(c) Now we want to evaluate the n − 1 dimensional integral in the previous part. Our strategy is to choose a particular f for which we can evaluate both sides of (2) explicitly and then deduce the value of the integral. The magic f is f(t) = e^{−t}. Using this conclude that if a_{i} > 0, i = 1 … n, then

∫_{{t∈R^{n−1}:t_{1}+...t_{n−1}<1,t_{i}>0}}(1−t_{1}−t_{2}−…−t_{n−1})^{a_{n−1}}t_{1}^{a_{1}−1} … t_{n−1}^{a_{n−1}−1} dt_{1} … dt_{n−1}

Observe that the last part of this problem establishes the formula for the beta function used above.

3. Show that if g(x) = F (|x|), and g ≥ 0, then

∫_{R^{n}} g(x) dx = c_{n} ∫_{0}^{∞} F (t)t^{n−1} dt.

Hint: Write F (|x|) = F ∘ S(x_{1}^{2} + … x_{n}^{2}) where S(t) = \sqrt{t}. Consider the integral

∫_{{x_{i}>0,i=1...n}} F ∘ S(x_{1}^{2} + … + x_{n}^{2}) dx_{1} … dx_{n}

make the change of variables x_{i}^{2} = t_{i} and apply the previous problem. Your proof should give the value of c_{n}.