(Note: I asked this question a few days ago on [math.stackexchange](https://math.stackexchange.com/questions/44754/seeking-rationale-for-hadamards-finite-part-of-a-divergent-integral) but didn't get any responses. I've therefore decided to post it here instead.)


I have a problem justifying throwing away the divergent term in order to obtain Hadamard's finite part. I find this step to be highly unusual, and it is not obvious to me how the resulting expression can be valid. I'd appreciate, if possible, intuitive arguments -- I assume there are some. Its not the mathematics that I have problems with at this stage. Its the intuition.


For example (taken from the Wikipedia [page](http://en.wikipedia.org/wiki/Hadamard_finite_part)) the finite part of the following integral


$$\int\_a^b \frac{f(t)}{(t-x)^2}\, dt = \lim\_{\varepsilon \to 0} \left[ \int\_a^{x-\varepsilon}\frac{f(t)}{(t-x)^2}\,dt + \int\_{x+\varepsilon}^b\frac{f(t)}{(t-x)^2}\,dt -\frac{2f(x)}{\varepsilon} \right]$$


involves throwing away the term $\frac{2f(x)}{\varepsilon}$. I find it hard to justify this step especially when the term is neither finite nor negligible.



Rationale for Hadamard's finite part of a divergent integral
This can be viewed as a meromorphic continuation in s of the distribution which is integration against |x-t|^s in t. M. Riesz (1938)first observed that this is a meromorphic continuation of convergent integrals, Gelfand-Shilov (1958) formalized this in the context of Schwartz' distributions. Gelfand-Graev's volume I discusses many such examples. The exponent -2 is not at the boundary of the region re(s)>-1 where the integral converges (absolutely), so the integral does not have an obvious, natural evaluation as a limit.


The seemingly whimsical discarding of the 2f(x)/epsilon term is merely an obscured version of a correct computation of the meromorphic continuation. 


Meromorphic continuation is one way to ensure uniqueness of a 'regularization' procedure, which otherwise may be ambiguous, or accidentally fail to have verifiable continuity and other properties.


