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$$\int_0^\infty\frac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$$ What kind of contour is suitable for this integral?

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If $a$ is a constant and $x$ is a constant and $b$ is the variable of integration, that seems like a poor choice of names, and makes it hard to read. In particular, $dx$ is very common, but I've never seen $db$ used in an integral. You should switch all $b$ and $x$ in the question... – Thomas Andrews Aug 22 '12 at 18:35
None: this integral has a non-integrable singularity at $b=0$ (unless $k=0$). – Robert Israel Aug 22 '12 at 18:43
It has simple poles(singularity points) at $b=0, -a/2 \pm \sqrt{a^2-4l}/2 $. To solve such integrals, we take help of contour integration, and residue theorem. I am not able to find proper contour to solve this integration. – Bandhu Aug 23 '12 at 11:42
@Bandhu: you are mistaken. The residue theorem applies to poles inside contours. The only time you see a pole on a contour is when it is eventually removable (e.g., $\sin{x}/x$). You might be interested in a principal value, but for this integral, it also makes no sense because your pole is at an endpoint of the integration region. In short, this integral is a no-go. – Ron Gordon Apr 23 '13 at 21:28

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