MathDB

Consider a function real-valued function with

C^{\infty}

-class on

\mathbb{R}

such that:
(a)

f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.

(b) For

x\neq 0,\ f(x)>0.

Judge whether the following integrals

(i),\ (ii)

converge or diverge, justify your answer.

(i)

\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.

(ii)

\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.

2010 Kyoto University, Master Course in Mathematics

Problem Statement