The Finite Part of Divergent Integrals with Logarithmic Factors

The concepts of the finite part (f.p.) and analytic finite part (a.f.p.) of divergent integrals are defined in the situation where the singular function in the integral has a logarithmic factor. The change of variables in f.p.- and a.f.p-integrals is examined.


Introduction
Divergent integrals and equations containing them have been studied for a long time, including the principal work done by Hadamard in [5]. Equations containing divergent integrals have been useful in applications in mathematics [1,6,7,10] and physics [8,9]. One of the concepts under discussion has been the summability, i.e. finding the finite part (f.p.), of divergent integrals. Over the years numerous approaches to defining the finite part of divergent integrals have been examined (see [5,7]). For methods on numerically finding the finite part of divergent integrals see [2,3,4]. In [11], a unified approach to summability of divergent integrals is presented that covers the approaches considered before. Later, in [12] summability methods for divergent integrals are studied in more detail in the case where the integrand is represented as a product of two functions, one with a parameter-dependent non-integrable singularity at one point of integration and the other absolutely integrable. Under discussion are methods which are based on the expansion of the absolutely integrable function in a Taylor series with center at the singular point (f.p.) and on the analytic continuation with respect to the parameter of the singularity (a.f.p.). Formulae of changes of variables in such integrals are also presented.

K. Lätt
More precisely, in [12], divergent integrals where C m,α [0, R] is the class of functions satisfying the conditions a ∈ C m [0, R], For λ ∈ C with Re λ < m + α the finite part of (1.1) is defined in terms of Taylor expansions: if λ ∈ C\N 0 , then These definitions have two crucial consequences. Firstly, the finite part integral defined by (1.3) is the analytic continuation of integral (1.1) from Re λ < 0 into For the change of variables r = g(ρ) with the following result is established in [12].
where a * (ρ, λ) = a(g(ρ)) (g(ρ)/ρ) −λ−1 g (ρ) and The analytic finite part of integrals R 0 a(r, λ)r −λ−1 dr is also defined. Here a(r, λ), with regards to λ ∈ C, is an analytic function. Moreover, the formula for change of variables in a.f.p.-integrals is given: . The first goal of the present paper is to define the finite part of integral R 0 a(r)r −λ−1 (ln r) n dr (1.5) with n ∈ N 0 so that the f.p.-integrals have the same two crucial properties they had in [12]. We will also show that a similar result to Theorem 1 holds for change of variables in the f.p.-integrals of (1.5). In the last part of the present article we define the analytic finite part of integral R 0 a(r, λ)r −λ−1 (ln r) n dr and examine the change of variables in a.f.p.-integrals. As it turns out, the logarithmic factor (ln r) n greatly complicates the situation. For instance, the correction term Π * (λ) in the formula for the change of variables now contains n + 1 terms (see Theorem 2).
Let λ ∈ C, Re λ < m + α. We define the finite part of (1.5) in the following way: for λ ∈ C\N 0 Both definitions (2.3) and (2.4) are based on the expansion of the absolutely integrable function a in a Taylor series with centre at the singular point (cf. [11]). To demonstrate that the integrals on the right-hand sides of (2.3) and (2.4) converge absolutely for Re λ < m + α, we show, similarly to the way it was done in [12], that (2.2) leads to If m = 0, then (2.5) is obvious. For m 1, by Taylor's formula with integral remainder term we have Now (2.5) follows from (2.2) and (2.6).
As mentioned before, one of the goals of this article was to define the finite part of integral (1.5) so that f.p.-integrals would have the same two crucial properties they had in article [12]. To be more precise, for λ ∈ C\N 0 , the f.p.-integral defined by (2.3) should be the analytic continuation of (1.5), and for points λ = l ∈ N 0 , a certain limit relation should hold. We close out this section by showing that with (2.3) and (2.4) we have achieved our goals.

The Analytic Finite Part of a Divergent Integral
Let us now examine integrals R 0 a(r, λ)r −λ−1 (ln r) n dr, (4.1) where n ∈ N 0 and a(r, λ) satisfies the following conditions: (AN1) for fixed r ∈ [0, R] the coefficient a(r, λ), as a function of λ, is analytic; (AN2) for fixed λ ∈ C the coefficient a(r, λ), as a function of r belongs to C m,α [0, R], and ∂ ∂r m a(r, λ) − ∂ ∂r m a(r, λ) r=0 M λ r α , 0 r R with an M λ that is bounded on bounded subsets of C.
We define the analytic finite part of divergent integral (4.1) as follows: for λ ∈ C\N 0 , Re λ < m + α a.f.p.   with l ∈ N 0 , l < m + α. To demonstrate the last equality we use formula (4.2) for points λ ∈ C\N 0 on the right-hand side of (4.3) and note that by (AN1) the integral and the terms in the corresponding sum with k = l are analytic functions at λ = l. Thus (cf. (2.8)) a.f.p.
To conclude, we show that, as in article [12], the biggest advantage of a.f.p. concept over f.p. is that the formula for change of variables does not contain a correction term. That is, the following result holds for change of variables in a.f.p.-integrals.