# Finite part integral for terms involving absolute value

Can the simple "recipe" for Cauchy principle value and Hadamard Finite Part be extended to find the finite part of integrals of the form $$\int_a^b f\big(\left|x-c\right|\big) \ dx$$ with $c \in [a,b]$ and $f(\left|x\right|) \sim \frac{1}{\left|x\right|} \ x \to 0$?

The most obvious extension seems to be $$\int_a^b f(\left|x-c\right|) \ dx = \int_a^{c-\epsilon} f\big(\left|x-c\right|\big) \ dx + \int_{c+\epsilon}^b f\big(\left|x-c\right|\big) \ dx + 2 \log{\epsilon} \lim_{x->c}\big(f\big(\left|x-c\right|\big)\left|x-c\right|\big)$$ but I'm not convinced that this 'recipe' can be justified by meromorphic continuation the way that the Hadamard case can. I don't know of any way to apply contour integration here due to the absolute value.

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