Contour integral representation of the finite part integral using Exponential integral

Authors

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

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Article ID

SPP-2019-2D-06

Section

Theoretical and Mathematical Physics

Published

2019-05-23

How to Cite

[1]
KJM Remo and EA 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://proceedings.spp-online.org/article/view/SPP-2019-2D-06.