# Many-valuedness of the finite part integral

### Abstract

We considered the finite parts of divergent integrals of the form ∫_{a}^{b}*g*(*x*) (*x*-*x*_{0})^{-n }d*x* for *a*<*x*_{0}<*b*. It is shown that the finite part takes on a broad range of values, depending on how the contribution of the offending singularity was removed. A conditionally convergent logarithmic term appeared for cases in which the two radii of the contours used possessed a linear dependence towards each other, proving that the finite part could take on many values. The multi-valued finite parts were finally expressed as contour integral representations in the complex plane.

*Proceedings of the Samahang Pisika ng Pilipinas*

**36**, SPP-2018-PC-38 (2018). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2018-PC-38.