I want to ask about the finitess of the following integral

\begin{equation} \int^1_0\frac{(1-s)^\alpha}{s^\beta}ds \end{equation}

when $\alpha>\beta>1$. This integral is very similar to the Beta function, aside from the negative power $-\beta$ here. Can anyone confirm if there really exists some big $\alpha$ such that the integral is finite? Thank you.

  • $\begingroup$ For $\beta \geqslant 1$, you have a non-integrable singularity at $0$. $(1-s)^\alpha$ is essentially $1$ there, so you are left with $$\int_0^\varepsilon \frac{ds}{s^\beta}.$$ $\endgroup$ – Daniel Fischer Feb 14 '14 at 21:25
  • $\begingroup$ Check $\beta$ function. $\endgroup$ – Mhenni Benghorbal Feb 14 '14 at 21:26
  • $\begingroup$ @canis89: You are welcome. $\endgroup$ – Mhenni Benghorbal Feb 14 '14 at 21:28

The integral is finite if and only if $$\beta<1\quad\text{and}\quad -\alpha<1\iff \beta<1\quad\text{and}\quad \alpha>-1$$

  • $\begingroup$ You're right, thank you. It's too much to hope for that this is finite. $\endgroup$ – canis89 Feb 14 '14 at 21:26
  • $\begingroup$ You're welcome. $\endgroup$ – user63181 Feb 14 '14 at 21:27
  • $\begingroup$ Nice work, Sami, as usual! $\endgroup$ – Namaste Feb 15 '14 at 13:35

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