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This should be a pretty simple question. I've just started learning about principal value integrals. But, I'm a little confused about poles.

There is an identity for the cauchy principal value integral. There are two actually, one is for a single pole in the interval of integration, and the other is for a double pole in the interval of integration. I can't seem to find any useful information on poles by searching google. Can someone please explain what a pole in the interval of integration is, and explain how to determine if an equation has a single or double pole?

Thank you to anyone that replies.

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up vote 3 down vote accepted

If $\lim_{z\to a}f(z)$ doesn't exist, but $\lim_{z\to a}(z-a)f(z)$ does, then $f$ has a single pole at $a$. If neither limit exists, but $\lim_{z\to a}(z-a)^2f(z)$ does, then $f$ has a double pole at $a$.

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Thank you for replying! – Silmaril89 Jun 16 '11 at 15:58

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