The finite part of a divergent integral, also called the Hadamard finite part or partie finie, is the finite value obtained by removing the divergent terms from the asymptotic expansion of a specified cutoff integral and taking the remaining limit (Galapon 2016). The cutoff and subtraction prescription are part of the definition.
For a simple pole in the interior of the integration interval, the finite part obtained
by symmetric exclusion of the pole agrees with the Cauchy
principal value. The two notions differ in general. For example, symmetric exclusion
about 
gives
 |
for 
. Since 
as 
, the Cauchy
principal value does not exist as a finite number. Removing the divergent term
instead gives
![FPint_(-1)^1(dx)/(x^2)=lim_(epsilon->0^+)[I(epsilon)-2/epsilon]=-2](/images/equations/FinitePart/NumberedEquation2.svg) |
(M. Onofri, pers. comm., Aug. 23, 2026).
The analogous prescription for partial sums gives a Hadamard-regularized sum.
