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SUMMARY

The integral \(\int_{p>0}^{\infty}\frac{\exp[-ap]}{p}dp\) diverges due to the behavior of the integrand. Specifically, the term \(e^{-ap}\) is continuous and bounded on the interval \(]0, 1]\), leading to a logarithmic divergence as \(\int_0^1 dp/p\). Additionally, the integrand remains positive on the interval \(]1, \infty[\), confirming that the overall result diverges. This conclusion is supported by the analysis of the integral's behavior across its defined limits.

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Yiping
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I try to find the solution of the integrals below for more than a week.
\int_{p>0}^{\infty}\frac{\exp[-ap]}{p}dp
can anyone give me a hint about how to solve it?
 
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e^{-a p} is continuous on ]0, 1] and therefore bounded (it lies between m and M). So you can estimate it to this constant and then on ]0, 1] the integral diverges logarithmically (\displaystyle\propto \int_0^1 dp/p). Furthermore, the integrand is positive on ]1, infinity[ so the end result also diverges.

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That is definitely true...
 

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