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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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