where is an analytic function on such that and as , is divergent. However, we can assign a finite value to this divergent integral. In fact, for , we have by integration by part
and the limit
exists and is finite. We call this limit an Hadamard finite-part (f.p.) integral and denote it by
More generally, we can define the f.p. integral
for , and an analytic function on such that and as .
In this paper, we propose a numerical method of computing a f.p. integral (1). In the proposed method, we express the desired f.p. integral using a complex integral, and obtain the f.p. integral by evaluating the complex integral by the DE formula . Theoretical error estimate and some numerical examples show that the proposed approximation formula converges exponentially as the number of sampling points increases.
Previous studies related to this paper are as follows. The author and Hirayama proposed a numerical method of computing ordinary integrals based on hyperfunction theory, a theory of generalized functions based on complex function theory . In their method, we obtain the desired integral by evaluating a complex integral using a conventional numerical integration formula as in the method for computing f.p. integrals proposed in this paper. The author proposed numerical methods of computing a f.p. integral with a singularity at an endpoint on a finite interval [6, 7] and a f.p. integral with an integral power singularity at the endpoint on a half infinite interval . Also in these methods, we obtain a desired integral using a complex integral, and obtain the integral by evaluating the complex integral by a conventional numerical integration formula. For the computation of a Cauchy principal value integral or a f.p. integral on a finite interval with a singularity in the interior of the integral interval
many numerical methods were proposed. Elliot and Paget proposed Gauss-type numerical integration formulas for (2) [1, 9]. Bialecki proposed Sinc numerical integration formula of computing (2), where the trapezoidal formula together with a variable transform technique are used as in the DE formula . The author et al. improved these methods and proposed a DE-type numerical integration formula of computing (2) .
The remainder of this paper is structured as follows. In Section 2, we define the f.p. integral (1) and propose a numerical method of computing it. In addition, we show a theoretical error estimate which shows the exponential convergence 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 finite-part integral and a numerical method
Let , , and be an analytic function on such that and as . The Hadamard finite-part integral (1) is defined by
We can show that (3) is well-defined as follows. In fact, by integration by part, we have for
Then, we have
and the limit of (3) exists and is finite.
As shown in the following theorem, a f.p. integral (3) is expressed using a complex integral, which is the bases of our numerical method.
We suppose that is analytic in a domain containing the positive real axis in its interior. Then, we have
where is a complex integral path such that it encircles the positive real axis in the positive sense and it is contained in , and is the principal value, that is, the branch such that it takes a real value on the positive real axis222The complex integral on the right-hand side of (4) coincides with the integral of as a hyperfunction . .
Proof of Theorem 1
By Cauchy’s integral theorem, we have
where , and and are the complex integral paths respectively given by
(see Figure 1).
Regarding the integrals on , we have
Regarding the integral on , we have
where we exchanged the order of the integration and the infinite sum on the fourth equality since the infinite sum is uniformly convergent on . Summarizing the above calculations, we have
and, taking the limit , we obtain (4).
where is the DE transform
We can take the positive integers small for a given mesh since the transformed integrand decays double exponentially as . Taking a parameterization of the integral path
and evaluating the complex integral of (4) by the DE formula, we obtain the approximation formula of the f.p. integral
A theoretical error estimate of the approximation (7) is given in the following theorem, where are taken to be for the simplicity.
We suppose that
is an analytic function in the strip
and the domain
is contained in ,
there exists positive numbers and such that
Then, we have the inequality
and is a positive number depending on and only.
This theorem shows that the proposed approximation (7) converges exponentially as the mesh decreases and the number of sampling points increases.
Proof of Theorem 2
Regarding the second term, we have
Therefore, we have (8).
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. Then, taking the integral path to be symmetric with respect to the real axis, that is,
which leads to , we have
3 Numerical examples
We computed the f.p. integrals
with by the proposed approximation formula (10). We performed all the computations using programs coded in C++ with double precision working. We took the complex integral path as
Figure 3 shows the relative errors of the proposed method (10) applied to the f.p. integrals (11). From these figures, the proposed formula converges exponentially as the number of sampling points increases.
In this paper, we proposed a numerical method of computing a f.p. integral with a non-integral power singularity at the endpoint over a half infinite interval. In the proposed method, we express the desired f.p. integral by a complex integral, and we obtain the f.p. integral by evaluating the complex integral by the DE formula. Theoretical error estimate and some numerical examples show that the proposed approximation converges exponentially as the mesh of the DE formula decreases and the number of sampling points increases for an analytic integrand.
The complex integral which expresses a desired f.p. integral and gives the basis of the proposed method coincides with the definition of the integral of a hyperfunction, a generalized function given by an analytic function. In hyperfunction theory , a hyperfunction is described by an analytic function called a defining function, and its integral is defined by an complex integral involving the defining function. In addition, in hyperfunction theory, we can deal with an ordinary integral and a f.p. integral in a unified way. We found the proposed method, that is, the method of computing a f.p. integral by evaluating a complex integral, from this viewpoint in hyperfunction theory. Therefore, we expect that hyperfunction theory is applicable to many numerical computations in science and engineering.
-  (1979) Gauss type quadrature rules for cauchy principal value integrals. Math. Comput. 33, pp. 301–309. External Links: Cited by: §1.
-  (1989) Regularization, pseudofunction, and hadamard finite part. J. Math. Anal. Appl. 141, pp. 195–207. External Links: Cited by: §1.
-  (2010) Introduction to hyperfunctions and their integral transforms — an applied and computational approach. Birkha̋user, Basel. Cited by: §4, footnote 2.
-  (2018) Numerical integration based on hyperfunction theory. J. Comput. Appl. Math. 327, pp. 243–259. Cited by: §1.
-  (2000) DE-type quadrature formulae for cauchy principal-value integrals and for hadamard finite-part itnegrals. In Proceedings of the Second ISAAC Congress, Vol. 1, pp. 357–366. Cited by: §1.
-  (2019) A numerical method for computing hadamard finite-part integrals with a non-integral power singularity at an endpoint. Note: arXiv:1909.11398v1 [math.NA] Cited by: §1.
-  (2019) A numerical method for hadamard finite-part integrals with an integral power singularity at an endpoint. Note: arXiv:1909.08872v1 [math.NA] Cited by: §1.
-  (2019) A numerical method of computing hadamard finite-part integrals with an integer power singularity at the endpoint on a half infinite interval. Note: arXiv:1910.00807v1 [math.NA] Cited by: §1.
-  (1981) The numerical evaluation of hadamard finite-part integrals. Numer. Math. 36, pp. 447–453. External Links: Cited by: §1.
-  (1993) Numerical methods based on sinc and analytic functions. Springer-Verlag, New York. Cited by: §2.
-  (1978) Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci., Kyoto Univ. 339, pp. 721–741. Cited by: §1, §1, §2.