2
$\begingroup$

Below equation is satisfied.

$ \int_{0}^{\infty} x^nf(x)dx=0 $

If $n$ is integer with $n\geq0$, then we can't guarantee $f(x) = 0$ for all positive $x$.

When $n$ is rational number with $n\geq0$ , do we get same result?

If so, what about $n$ is real number with $n\geq 0$?

Above integral is improper Riemann integral.

I forgot one condition. f(x) is continuous function.

  • 2
    What is the meaning of $\int_0^\infty x^n f(x)\,dx$: is it the Lebesgue integral or improper Riemann integral?2012-09-07

0 Answers 0