1 Introduction
The integral
where is an analytic function on the closed interval , is divergent. However, we can assign a finite value to this divergent integral as follows. For , using integration by part, we have
and the limit
is finite. We call this limit an Hadamard finitepart (f.p.) integral and denote it by
Similarly, we can define a f.p. integral
(1) 
for , and an analytic function on [6].
In this paper, we propose a numerical method of computing f.p. integrals (1). In the proposed method, we express the f.p. integral using a complex loop integral, and we obtain the desired f.p. integral by evaluating the complex integral by the trapezoidal formula with equal mesh. Theoretical error estimate and numerical examples will show that the approximation by the proposed method converges exponentially as the number of sampling points increases.
Previous works related to this paper are as follows. The author and Hirayama proposed a numerical integration method for ordinary integrals related to hyperfunction theory [8], where a desired integral is expressed using a complex loop integral, and it is obtained by evaluating the complex integral by the trapezoidal formula with equal mesh. The author proposed a numerical method of computing f.p. integrals with an integral order singularity [10]. For Cauchy principal value integrals and Hadamard finitepart integrals with a singularity in the interior of the integral interval
(2) 
many numerical methods were proposed. Elliot and Paget proposed a Gausstype numerical integration formulas for f.p. integrals (2) [5, 11]. Bialecki proposed Sinc numerical integration formulas for f.p. integrals [3, 2], where the trapezoidal formula with the variable transform technique are used as in the DE formula for ordinary integrals [12]. The author et al. improved them and proposed a DEtype numerical integration formulas for f.p. integrals (2) [9].
The remainder of this paper is structured as follows. In Section 2, we define the f.p. integrals and propose a numerical method of computing them. Then, we give a theorem on error estimate of the proposed method. In Section 3, we show some numerical examples which show the effectiveness of the proposed method. In Section 4, we give a summary of this paper.
2 Hadamard finitepart integral
The Hadamard finitepart integral is defined by
(3) 
We can see that it is welldefined using integration by part. In fact, repeating integration by part, we have
If the integrand is an analytic function on the closed interval , the f.p. integral (3) is expressed using a complex loop integral as in the following theorem.
Theorem 1
We suppose that is an analytic function in a complex domain containing the closed interval in its interior. Then, the f.p. integral (3) is expressed as
(4)  
where  
(5) 
and is a closed complex integral path contained in and encircling the interval in the positive sense.
Proof of Theorem 1
From Cauchy’s integral theorem, the complex integral of the first term on the righthand side of (4) is modified into
where , , and are complex paths respectively defined by
with (see Figure 1).
From the formula 15.3.7 in [1], we have
Then, as to the integrals on , we have
where we remark that is a singlevalued analytic function on the interval . As to the integral on , we have
The first term on the righthand side is written as
and the second term is written as
(the second term)  
As to the integral on , from the formula 15.3.10 in [1], we have
where is the Digamma function: and  
and, then, the integral on is of . Summarizing the above calculations, we have
and, taking the limit , we have (4).
We can obtain the desired f.p. integral (3) by evaluating the complex integral in (4) on the closed integral path , which is parameterized by , , by the trapezoidal formula with equal mesh as follows.
(6) 
The hypergeometric function in the definition of in (5) is easily evaluated using the continued fraction expansion (see §12.5 in [7]). If is an analytic curve, the complex loop integral is an integral of an analytic periodic function on an interval of one period, to which the trapezoidal formula with equal mesh is very effective, and, then, the approximation formula (6) is very accurate. In fact, applying the theorem in §4.6.5 in [4] to the approximation of the complex integral in (6), we have the following theorem on error estimate of the proposed approximation.
Theorem 2
We suppose that

the parameterization function of is analytic in the strip domain

the domain
is contained in , and

the function is analytic in .
Then, we have for arbitrary
(7)  
where  
(8) 
This theorem says that the proposed approximation (6) converges exponentially as the number of sampling points increases if is an analytic periodic function and is an analytic curve.
We remark here that, if is real valued on the real axis, we can reduce the number of sampling points by half. In fact, in this case, we have
from the reflection principle, and, taking the integral path to be symmetric with respect to the real axis, that is,
we have
(9) 
3 Numerical examples
In this section, we show some numerical examples which show the effectiveness of the proposed method. We computed the integrals
(10)  
with by the approximation formula (9). All the computations were performed using programs coded in C++ with double precision working. The complex integral path was taken as the ellipse
with for the integral (i) and for the integral (ii). Figure 2 show the relative errors of the proposed method applied to the integrals (i) and (ii) as functions of the number of sampling points . From these figures, the errors of the proposed method decays exponentially as increases, and the decay rate of the error does not depend much on . Table 1 shows the decay rates of the errors of the proposed method applied to the f.p. integrals (i) and (ii).
(i)  (ii) 
1  2  3  4  

integral (1)  
integral (2) 
4 Summary
In this paper, we proposed a numerical method of computing Hadamard finite part integrals with a nonintegral power singularity on an endpoint. In the proposed method, we express the desired f.p. integral using a complex loop integral, and obtain the f.p. integral by evaluating the complex integral by the trapezoidal formula with equal mesh. Theoretical error estimate and some numerical examples showed the exponential convergence of the proposed method.
We can obtain similarly f.p. integrals on an infinite interval. This will be reported in a future paper.
References
 [1] (1965) Handbook of mathematical functions with formulas, graphs and mathematical tables. Dover, New York. Cited by: §2.
 [2] (1990) A sinc quadrature rule for hadamard finitepart integrals. Numer. Math. 57, pp. 263–269. External Links: Document Cited by: §1.
 [3] (1990) A sinchunter quadrature rule for cauchy principal value integrals. Math. Comput. 55, pp. 665–681. External Links: Document Cited by: §1.
 [4] (1984) Methods of numerical integration, second ed.. Academic Press, San Diego. Cited by: §2.
 [5] (1979) Gauss type quadrature rules for cauchy principal value integrals. Math. Comput. 33, pp. 301–309. External Links: Document Cited by: §1.
 [6] (1989) Regularization, pseudofunction, and hadamard finite part. J. Math. Anal. Appl. 141, pp. 195–207. External Links: Document Cited by: §1.
 [7] (1977) Applied and computational complex analysis. Vol. 2, John Wiley & Sons, New York. Cited by: §2.
 [8] (2018) Numerical integration based on hyperfunction theory. J. Comput. Appl. Math. 327, pp. 243–259. Cited by: §1.
 [9] (2000) DEtype quadrature formulae for cauchy principalvalue integrals and for hadamard finitepart itnegrals. In Proceedings of the Second ISAAC Congress, Vol. 1, pp. 357–366. Cited by: §1.
 [10] (2019) A numerical method for hadamard finitepart integrals with an integral power singularity at an endpoint. Note: arXiv:1909.08872v1 [math.NA] Cited by: §1.
 [11] (1981) The numerical evaluation of hadamard finitepart integrals. Numer. Math. 36, pp. 447–453. External Links: Document Cited by: §1.
 [12] (1978) Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci., Kyoto Univ. 339, pp. 721–741. Cited by: §1.