Contour integral representation of the finite part integral using Exponential integral

Abstract

Finite part integral is one way of assigning meaningful values to a divergent integral of the form ∫₀^a f(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.