Contour integral representation of the finite part integral using Exponential integral

  • Kenneth Jhon Mora Remo Theoretical Physics Group, National Institute of Physics, University of the Philippines Diliman
  • Eric A. Galapon National Institute of Physics, University of the Philippines Diliman

Abstract

Finite part integral is one way of assigning meaningful values to a divergent integral of the form ∫₀f(x) x⁻(m+ν)dx. In the most common way, it is obtained by evaluating it in the real line. However, its definition has been extended in the complex plane as shown in [E. A. Galapon RSPA 473, 20160567 (2016)]. The complex extension is done by representing it using contour integral. The representation is dependent on the value of ν. Here, we considered the case when ν=0, and obtained a new contour integral representation of the finite part with the use of Ei(-z) as our basis function. We also obtained a term by term integration of the incomplete Stieltjes transform which turned out to be consistent with the results shown in the paper discussed above.

Published
2019-05-23
How to Cite
[1]
K. J. Remo and E. Galapon. Contour integral representation of the finite part integral using Exponential integral, Proceedings of the Samahang Pisika ng Pilipinas 37, SPP-2019-2D-06 (2019). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2019-2D-06.
Section
Theoretical and Mathematical Physics