Find upper limit of improper integral - Numerical Integration

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The discussion centers on finding the upper limit of the improper integral defined as \(\int _0^{\infty }x\frac{\beta^\alpha}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\beta/x}dx\). The integral converges to \(\frac{(1/\beta)^\alpha \beta^{\alpha+1}}{(\alpha-1)}\). Alex seeks methodologies for deriving the upper bound for numerical evaluation. A suggested resource for further exploration is provided, linking to a relevant method discussed on Physics Forums.

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Find upper limit of improper integral -- Numerical Integration

Hi,

I have the following complicated integral willing to integrate numerically. The integral is:

[itex]\int _0^{\infty }x\frac{\beta^\alpha}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\beta/x}dx[/itex].

We know that the integral converges to:
[itex]((1/\beta)^\alpha \beta^{\alpha+1})/(\alpha-1)[/itex].

Can anyone suggest a methodology of deriving the upper bound of the limit of this integral for numerical evaluation purposes?

Thanks for your time.

BR,

Alex
 
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