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I am learning about cauchy principal value integrals and was looking for help evaluating the integral given by

\begin{align} \lim_{\epsilon \to 0} \int_{-a}^{-\epsilon} \dfrac{A}{x} dx + \lim_{\epsilon \to 0}\int_{\epsilon}^a \dfrac{B}{x} dx \end{align}

where $A$ and $B$ are constants. My attempt so far is:

\begin{align} \lim_{\epsilon \to 0} \int_{-a}^{-\epsilon} \dfrac{A}{x} dx + \lim_{\epsilon \to 0}\int_{\epsilon}^a \dfrac{B}{x} dx = \lim_{\epsilon \to 0} [A(\ln \epsilon - \ln a)+B(\ln a - \ln \epsilon )] \end{align}

If $A = B$, then I can cancel out the $\ln \epsilon$ terms and the principal value is 0. However, I would like to know how to determine the principal value if $A \neq B$.

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    It is rather unusual to look for the PV (Principal Value) of a function that is different on the left and on the right of the singularity. – 2017-02-10
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    From my understanding, Cauchy's principal value can sometimes be found using the residue theorem. In cases where we use Jordan's lemma and find the residue around an arc with angle $\phi$, wouldn't we encounter integrals that are different along the left and right side of the singularity? – 2017-02-10

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If $A > B$ the the principal value is $-\infty$. If $A < B$ then the principal value is $+\infty$.