MHB How can i integrate this function

[Another1]

\[ \int_{0}^{\inf} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\] or \[ \int_{0}^{constant} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\]

maybe application Residue theorem integral ? because this problem same the kramers kronig relation?

 

[HOI]

The "residue theorem" applies to functions of a complex variable. Are you planning to rewrite the function as a function of a complex variable, with path of integration the x-axis from x- R to x+ R and the semicircle $z= R \cos(\theta)+ iR \sin(\theta)4$ and then take R going to infinity?

If b and c are real numbers then you path will need to loop around b and c on the real line. Whether your loops include b and c or not you will need to calculate the integral over those small semi-circles.