1

Let $f:[0,\infty [\to \mathbb R$ a continuous function such that $\int_1^\infty \frac{f(x)}{x}dx$ converge. Prove that $$\int_0^\infty \frac{f(ax)-f(bx)}{x}dx$$ converge and compute this integral.

Attempts

For the convergence : Since for all $t>0$, $$\int_t^\infty \frac{f(ax)}{x}dx\quad \text{and}\quad \int_t^\infty \frac{f(bx)}{x}dx$$

converge, we have that $\int_0^\infty \frac{f(ax)}{x}dx$ and $\int_0^\infty \frac{f(bx)}{x}dx$ that converge, and thus $$\int_0^\infty \frac{f(ax)-f(bx)}{x}dx.$$

But how I can compute this integral ?

Peter
  • 1,005
  • 1
    Your argument for convergence is not correct. Example : $\int_t^\infty \frac{1}{x^2}dx$, $t>0$. – Surb Jul 16 '18 at 17:34
  • 1
    That's a Frullani's integral. You may read about them https://math.stackexchange.com/questions/61828/proof-of-frullanis-theorem – openspace Jul 16 '18 at 17:37

0 Answers0