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I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that

$$ \int_0^{\epsilon p}\exp\left(-x\cdot\tfrac{1}{p}\right)\,\,\mathrm{d}x = p(1-e^{-\epsilon}) $$

I am interested in calculating the related integral

$$ \int_0^{\epsilon p}\exp\left(-x\cdot\tfrac{1-\delta x}{p-\delta x}\right)\,\mathrm{d}x. $$

I have tried Mathematica to no avail. I would appreciate any help. I would love to hear the reasoning behind the steps if someone has a solution.

Thanks a lot!

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  • $\begingroup$ Are you asking for a closed form or an asymptotic? $\endgroup$ – Igor Rivin Oct 13 '14 at 20:28
  • $\begingroup$ A closed form would be great, if one exists. If not, understanding asymptotic behavior would be very helpful. $\endgroup$ – Mert Sağlam Oct 13 '14 at 20:30
  • $\begingroup$ Have you tried steepest descent for asymptotics? Also are you interested in asymptotics for large $p$ or $\delta$ $\endgroup$ – Alex R. Oct 13 '14 at 20:35
  • $\begingroup$ Thanks, I am looking up steepest descent, I have not used it before. I am interested in the behavior as $\epsilon$ gets smaller. $\endgroup$ – Mert Sağlam Oct 13 '14 at 20:53
  • $\begingroup$ Mathematica gives the series expansion $p\epsilon+O(\epsilon^2)$. $\endgroup$ – Stopple Oct 13 '14 at 21:07

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