Limit using the Sohotski-Plemenj formula

[muzialis]

Hi All, I am desperate to understand a calculation presented in a paper by Sethna, "Elastic theory has zero radius of convergence", freely available online

$$ lim_{\epsilon \to +0}Z(-P+i\epsilon) = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{ -\alpha y -\beta x^2 - y(-P+i \epsilon)x \}$$

The author says use is to be made of the Theorem

$$ \lim_{\epsilon \to +0} \int \mathrm{d}x f(x)/(x+i \epsilon) = P.V. \int \mathrm{d}x f(x)/x -i \pi f(0) $$

but how to get there?

My best attempt is to integrate on the y variable, obtaining

$$ lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \exp \{ -\beta x^2 \} [- \frac{1}{-\alpha-(-P+i \epsilon)x} ]$$

To use the Theorem I need the denominator to be of the form y + i*epsilon, which is not as in the attempt I try the variable of Integration is multiplied by epsilon so I am unable to proceed.

I am even more puzzled because if I set epsilon to zero, the integral I wrote coincides with the P.V. integral reported in the paper as the P.V. contribution in the Sohotski-Plemelj formula.

How to calculate that limit then?

Any help would be so appreciated

 

[eljose79]

.Hello,

Thank you for reaching out for help with this calculation. The theorem mentioned in the paper is known as the Sokhotski-Plemelj formula, and it is a useful tool for calculating integrals with singularities. Here's how you can use it to calculate the limit in question:

First, let's rewrite the integral in a slightly different form:

$$ lim_{\epsilon \to +0}Z(-P+i\epsilon) = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{ -\alpha y -\beta x^2 - y(-P+i \epsilon)x \}$$

$$ = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{ -\beta x^2 \} \exp \{-\alpha y - y(-P+i \epsilon)x \} $$

Next, we can use the Sokhotski-Plemelj formula to rewrite the second exponential term in the integrand as a principal value integral:

$$ = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{ -\beta x^2 \} \left[ \frac{-1}{-\alpha-(-P+i \epsilon)x} \right] $$

$$ = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{ -\beta x^2 \} \left[ \frac{1}{\alpha+(P-i \epsilon)x} \right] $$

Now, we can use the theorem mentioned in the paper to calculate the limit of this integral:

$$ = P.V. \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{ -\beta