A numerical method of computing Hadamard finite-part integrals with an integral power singularity at the endpoint on a half infinite interval

1 Introduction

The integral

\int_{0}^{\infty} x^{- 1} f (x) d x,

where $f (x)$ is an analytic function on the half infinite interval $[0, + \infty)$ such that $f (0) \neq 0$ and $f (x) = O (x^{- α})$ with $α > 0$ , is divergent. However, we can assign a finite value to this divergent integral. In fact, we consider the integral

\int_{ϵ}^{\infty} x^{- 1} f (x) d x

with $ϵ > 0$ , and, by integrating by part, we have

	$\int_{ϵ}^{\infty} x^{- 1} f (x) d x =$	$\int_{ϵ}^{\infty} (log x)^{'} f (x) d x$
	$=$	$[log x f (x)]_{ϵ}^{\infty} - \int_{ϵ}^{\infty} log x f^{'} (x) d x$
	$=$	$- f (ϵ) log ϵ - \int_{ϵ}^{\infty} log x f^{'} (x) d x$
	$=$	$- f (0) log ϵ - \int_{0}^{\infty} log x f^{'} (x) d x + O (1) (as ϵ ↓ 0) .$

Then, the limit

lim ϵ ↓ 0 {\int_{ϵ}^{\infty} x^{- 1} f (x) d x + f (0) log ϵ}

exists and is finite. We call this limit the Hadamard finite-part (f.p.) integral and denote it by

f . p . \int_{0}^{\infty} x^{- 1} f (x) d x .

In general, we can define the f.p. integral

I^{(n)} [f] = f . p . \int_{0}^{\infty} x^{- n} f (x) d x (n = 1, 2, \dots),

(1)

where $f (x)$ is an analytic function such that $f (0) \neq 0$ and $f (x) = O (x^{n - α - 1})$ $(x \to + \infty)$ with $α > 0$ [4].

In this paper, we propose a numerical method of computing f.p. integrals (1). In the proposed method, we express the f.p. integral (1) using a complex integral, and we obtain the f.p. integral by evaluating the complex integral by the DE formula [11]. Theoretical error estimate and some numerical examples show the exponential convergence of the proposed formula as the number of sampling points increases.

Previous works related to this paper are as follows. The author and Hirayama proposed a numerical method of computing ordinary integrals related to hyperfunction theory [5], a theory of generalized functions based on complex function theory. The author proposed numerical methods for computing Hadamard finite-part integrals with a singularity at an endpoint on a finite interval [8, 7]. In these methods, we express a desired integral using a complex integral, we obtain the integral by evaluating the complex integral by conventional numerical integration formulas. For Cauchy principal-value integrals or Hadamard finite-part integrals on a finite interval with a singularity in the interior of the integral interval

f . p . \int_{0}^{1} \frac{f (x)}{(x - λ)^{n}} d x (0 < λ < 1, n = 1, 2, \dots),

(2)

many methods were proposed. Elliot and Paget proposed Gauss-type numerical integration formulas for (2) [3, 9]. Bialecki proposed Sinc numerical integration formulas for (2) [2, 1], where the trapezoidal formula is used together with variable transform technique as in the DE formula [11]. Ogata and et al. improved them and proposed a DE-type numerical integration formula for (2) [6].

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 theoretical error estimate of the proposed method. In Section 3, we show some numerical example which show the effectiveness of the proposed method. In Section 4, we give a summary of this paper.

2 Hadamard finite-part integrals and a numerical method

The f.p. integral (1) is defined by

(3)

where $f (x)$ is an analytic function on $[0, + \infty)$ such that $f (0) \neq 0$ and $f (x) = O (x^{n - 1 - α})$ as $x \to + \infty$ with $α > 0$ , and the second term on the right-hand side is zero if $n = 1$ . We can show that it is well-defined as follows. In fact, for $ϵ > 0$ , we can show by integrating by part

		$\int_{ϵ}^{\infty} x^{- n} f (x) d x$
	$=$	$\frac{ϵ^{1 - n}}{n - 1} f (ϵ) + \frac{1}{n - 1} \int_{ϵ}^{\infty} x^{1 - n} f^{'} (x) d x$
	$=$	$\frac{ϵ^{1 - n}}{n - 1} f (ϵ) + \frac{ϵ^{2 - n}}{(n - 1) (n - 2)} f^{'} (ϵ) + \frac{1}{(n - 1) (n - 2)} \int_{ϵ}^{\infty} x^{2 - n} f^{''} (x) d x$
	$=$	$\dots$
	$=$	$n - 2 \sum k = 0 \frac{ϵ^{k + 1 - n}}{(n - 1) (n - 2) \dots (n - 1 - k)} f^{(k)} (ϵ) - \frac{log ϵ}{(n - 1)!} f^{(n - 1)} (ϵ)$
		$+ \frac{1}{(n - 1)!} \int_{ϵ}^{\infty} log x f^{(n)} (x) d x$
	$=$	$n - 2 \sum k = 0 \frac{ϵ^{k + 1 - n}}{(n - 1) (n - 2) \dots (n - 1 - k)} {\infty \sum l = 0 \frac{f^{(k + l)} (0)}{l!} ϵ^{l}}$
		$- \frac{log ϵ}{(n - 1)!} \infty \sum k = 0 \frac{f^{(k)} (0)}{k!} ϵ^{k} + \frac{1}{(n - 1)!} \int_{ϵ}^{\infty} log x f^{(n)} (x) d x$
	$=$	$n - 2 \sum l = 0 {l \sum k = 0 \frac{1}{(l - k)! (n - 1) (n - 2) \dots (n - 1 - k)}      (*)} ϵ^{l + 1 - n} f^{(l)} (0)$
		$- \frac{log ϵ}{(n - 1)!} f^{(n - 1)} (0) + O (1) (as ϵ ↓ 0),$

and

	$(*) =$	$\frac{1}{l! (n - 1)} + \frac{1}{(l - 1)! (n - 1) (n - 2)}$
		$+ \dots + \frac{1}{1! (n - 1) (n - 2) \dots (n - l + 1) (n - l)}$
		$+ \frac{1}{(n - 1) (n - 2) \dots (n - l + 1) (n - l) (n - l - 1)}$
	$=$	$\frac{1}{l! (n - 1)} + \frac{1}{(l - 1)! (n - 1) (n - 2)} + \dots$
		$+ \frac{1}{2! (n - 1) (n - 2) \dots (n - l + 2) (n - l + 1)}$
		$+ \frac{1}{1! (n - 1) (n - 2) \dots (n - l + 2) (n - l + 1) (n - l - 1)}$
	$=$	$\frac{1}{l! (n - 1)} + \frac{1}{(l - 1)! (n - 1) (n - 2)} + \dots$
		$+ \frac{1}{3! (n - 1) (n - 2) \dots (n - l + 3) (n - l + 2)}$
		$+ \frac{1}{2! (n - 1) (n - 2) \dots (n - l + 3) (n - l + 2) (n - l - 1)}$
	$=$	$\dots$
	$=$	$\frac{1}{l! (n - l - 1)} .$

Then, we have

	$\int_{ϵ}^{\infty} x^{- n} f (x) d x =$	$n - 2 \sum l = 0 \frac{ϵ^{l + 1 - n}}{l! (n - l - 1)} f^{(l)} (0) - \frac{log ϵ}{(n - 1)!} f^{(n - 1)} (0)$
		$+ O (1) (as ϵ ↓ 0) .$

Therefore, the limit in (3) exists and is finite.

The f.p. integral (3) is expressed using a complex integral.

Theorem 1

We suppose that $f (z)$ is analytic in a complex domain $D$ , which contains the half infinite interval $[0, + \infty)$ in its interior. Then, the f.p. integral (3) is expressed as

I^{(n)} [f] = \frac{1}{2 π i} \oint_{C} z^{- n} f (z) log (- z) d z,

(4)

where $C$ is a complex integral path such that it encircles $[0, + \infty)$ in the positive sense and is contained in $D$ .

Proof of Theorem 1

From Cauchy’s integral theorem, we have

\frac{1}{2 π i} \oint_{C} z^{- n} f (z) log (- z) d z = \frac{1}{2 π i} (\int_{Γ_{ϵ}^{(+)}} + \int_{C_{ϵ}} + \int_{Γ_{ϵ}^{(-)}}) z^{- n} f (z) log (- z) d z,

for $ϵ > 0$ , where $Γ_{ϵ}^{(+)}$ and $C_{ϵ}$ are complex integral paths respectively defined by

	$Γ_{ϵ}^{(+)} =$	${x \in R + i 0 \| + \infty > x ≧ ϵ},$
	$Γ_{ϵ}^{(-)} =$	${x \in R - i 0 \| ϵ ≦ x < + \infty},$
	$C_{ϵ} =$	${ϵ e^{i θ} \| 0 ≦ θ ≦ 2 π}$

(see Figure 1), and the complex logarithmic function $log z$ is the principal value, that is, the branch such that it takes a real value on the positive real axis.

Figure 1: The complex integral paths $Γ_{ϵ}^{(\pm)}$ and $C_{ϵ}$ .

As to the integrals on $Γ_{ϵ}^{(\pm)}$ , we have

		$\frac{1}{2 π i} (\int_{Γ_{ϵ}^{(+)}} + \int_{Γ_{ϵ}^{(-)}}) z^{- n} f (z) log (- z) d z$
	$=$	$\frac{1}{2 π i} {- \int_{ϵ}^{\infty} x^{- n} f (x) log (x e^{- i π}) d x + \int_{ϵ}^{\infty} x^{- n} f (x) log (x e^{i π}) d x}$
	$=$	$\frac{1}{2 π i} {- \int_{ϵ}^{\infty} x^{- n} f (x) (log x - i π) d x + \int_{ϵ}^{\infty} x^{- n} f (x) (log x + i π) d x}$
	$=$	$\int_{ϵ}^{\infty} x^{- n} f (x) d x .$

As to the integral on $C_{ϵ}$ , we have

	$\frac{1}{2 π i} \int_{C_{ϵ}} z^{- n} f (z) log (- z) d z$
$=$	$\frac{1}{2 π i} \int_{0}^{2 π} (ϵ e^{i θ})^{- n} f (ϵ e^{i θ}) log (ϵ e^{i (θ - π)}) i ϵ e^{i θ} d θ$
$=$
$=$	$\frac{1}{2 π} \infty \sum k = 0 \frac{ϵ^{k - n + 1}}{k!} log ϵ f^{(k)} (0) \int_{0}^{2 π} e^{i (k - n + 1) θ} d θ$
	$+ \frac{i}{2 π} \infty \sum k = 0 \frac{ϵ^{k - n + 1}}{k!} f^{(k)} (0) \int_{0}^{2 π} (θ - π) e^{i (k - n + 1) θ} d θ,$	(5)

where we exchanged the order of the integral and the infinite sum since the infinite series is uniformly convergent on $0 ≦ θ ≦ 2 π$ . Since

	$\int_{0}^{2 π} e^{i (k - n + 1) θ} d θ = 2 π δ_{k, n - 1},$
	$\int_{0}^{2 π} (θ - π) e^{i (k - n + 1) θ} d θ = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{- 2 π}{i (n - 1 - k)} & (0 ≦ k ≦ n - 2) 0 & (k = n - 1), \end{matrix}$

we have

(???) = \frac{log ϵ}{(n - 1)!} f^{(n - 1)} (0) - n - 2 \sum k = 0 \frac{ϵ^{k - n + 1}}{k! (n - 1 - k)} f^{(k)} (0) + O (ϵ) (ϵ ↓ 0) .

Summarizing the above calculations, we have

	$\frac{1}{2 π i} \oint_{C} z^{- n} f (z) log (- z) d z =$	$\int_{ϵ}^{1} x^{- n} f (x) d x - n - 2 \sum k = 0 \frac{ϵ^{k - n + 1}}{k! (n - 1 - k)} f^{(k)} (0)$
		$+ \frac{log ϵ}{(n - 1)!} f^{(n - 1)} (0) + O (ϵ) (ϵ ↓ 0),$

and, taking the limit $ϵ ↓ 0$ , we obtain (4).

We obtain the f.p. integral by evaluating the complex integral on the right-hand side of (4) by a conventional numerical integration formula such as the DE formula [11], that is,

\int_{- \infty}^{\infty} g (u) d u = \int_{- \infty}^{\infty} g (ψ_{D E} (v)) ψ_{D E}^{'} (v) d v ≃ h N_{+} \sum k = - N_{-} g (ψ_{D E} (k h)) ψ_{D E}^{'} (k h),

where $h > 0$ is the mesh of the trapezoidal formula, $u = ψ_{D E} (v)$ is the DE transform

ψ_{D E} (v) = {\begin{matrix} sinh (sinh v) & (g (u) = O (| u |^{- α - 1}) as u \to \pm \infty, α > 0) sinh v & (g (u) = O (exp (- c | u |)) as u \to \pm \infty, c > 0), \end{matrix}

and $N_{\pm}$ is a positive integer such that the transformed integrand
$g (ψ_{D E} (k h)) ψ_{D E}^{'} (k h)$ is sufficiently small at $k = - N_{-}, N_{+}$ . We can take $N_{\pm}$ small since $g (ψ_{D E} (v)) ψ_{D E}^{'} (v)$ decays double exponentially as $v \to \pm \infty$ . Then, we have the approximation formula

$I^{(n)} [f] ≃$	$I_{h, N_{+}, N_{-}}^{(n)} [f]$
$\equiv$	$\frac{h}{2 π i} N_{+} \sum k = - N_{-} φ (ψ_{D E} (k h))^{- n} f (φ (ψ_{D E} (k h))) log (- φ (ψ_{D E} (k h)))$
	$\times φ^{'} (ψ_{D E} (k h)) ψ_{D E}^{'} (k h),$	(6)

where $z = φ (u), - \infty < u < + \infty$ is a parameterization of the complex integral path $C$ .

If $f (x)$ is an analytic function on the real axis and $C$ is an analytic curve, the proposed approximation (6) converges exponentially as shown in the following theorem. For the simplicity, we take $N_{+} = N_{-} \equiv N^{'}$ .

Theorem 2

We suppose that

the parameterization function $φ (w)$ of $C$ is analytic in the strip

$D_{d} = {w \in C | | Im w | < d} (d > 0)$

such that

$φ (D_{d}) = {φ (w) | w \in D_{d}},$

is contained in $C ∖ [0, + \infty]$ ,
$N (f, φ, ψ_{D E}, D_{d})$

$\equiv$ $lim ϵ \to 0 \oint_{\partial D_{d} (ϵ)} | φ (ψ_{D E} (w))^{- n} f (φ (ψ_{D E} (w))) log (- φ (ψ_{D E} (w))) ψ_{D E}^{'} (w) |$

$<$ $\infty,$

where

$D_{d} (ϵ) \equiv {w \in C | | Re w | < 1 / ϵ, | Im w | < d (1 - ϵ)} .$

and
there exist positive numbers $C_{0}$ , $c_{1}$ and $c_{2}$ such that

$| f (φ (ψ_{D E} (v))) | ≦ C_{0} exp (- c_{1} exp (c_{2} | v |)) (\forall v \in R) .$

Then, we have the inequality

	$\| I^{(n)} [f] - I_{h, N^{'}}^{(n)} [f] \| ≦$	$\frac{1}{2 π} N (f, φ, ψ_{D E}, D_{d}) \frac{exp (- 2 π d / h)}{1 - exp (- 2 π d / h)}$
		$+ C (f, φ, ψ_{D E}, D_{d}) exp (- c_{1} exp (c_{2} N^{'} h)),$		(7)

where $I_{h, N^{'}}^{(n)} [f] = I_{h, N^{'}, N^{'}}^{(n)} [f]$ and $C (f, φ, ψ_{D E}, D_{d})$ is a positive number depending on $f (z)$ , $φ$ , $ψ_{D E}$ and $D_{d}$ only.

This theorem shows that the approximation formula (6) converges exponentially as the mesh $h$ decreases and the number of sampling points $2 N^{'} + 1$ increases.

Proof of Theorem 2

We have

	$∣ ∣ ∣ f . p . \int_{0}^{\infty} x^{- n} f (x) d x - I_{h, N}^{(n)} [f] ∣ ∣ ∣$
$≦$	$∣ ∣ ∣ f . p . \int_{0}^{\infty} x^{- n} f (x) d x - I_{h}^{(n)} [f] ∣ ∣ ∣$
	$+ h \sum \| k \| > N ∣ ∣ φ (ψ_{D E} (k h))^{- n} f (φ (ψ_{D E}) (k h)) φ^{'} (ψ_{D E} (k h)) ψ_{D E}^{'} (k h) ∣ ∣,$	(8)

where $I_{h}^{(n)} [f] = {lim}_{N \to \infty} I_{h, N}^{(n)} [f]$ . For the first term on the right-hand side of (8), we have

∣ ∣ ∣ f . p . \int_{0}^{\infty} x^{- n} f (x) d x - I_{h}^{(n)} [f] ∣ ∣ ∣ ≦ \frac{1}{2 π} N (f, φ, ψ_{D E}, D_{d}) \frac{exp (- 2 π d / h)}{1 - exp (- 2 π d / h)}

by Theorem 3.2.1 in [10]. For the second term on the right-side hand, we have

	$\| the second term \| ≦$	$C_{0} h \sum \| k \| > N exp (- c_{1} exp (c_{2} k h))$
	$≦$	$2 C_{0} \int_{k h}^{\infty} exp (- c_{1} exp (c_{2} x)) d x$
	$≦$	$2 C_{0} \int_{k h}^{\infty} exp (c_{2} x) exp (- c_{1} exp (c_{2} x)) d x$
	$=$	$\frac{2 C_{0}}{c_{2}} exp (- c_{1} exp (c_{2} N h)) .$

Then, we obtain (7).

We remark here that we can reduce the number of sampling points by half if the integrand $f (x)$ is real valued on the real axis. In fact, we have $f (¯ ¯ ¯ z) = ¯ ¯¯¯¯¯¯¯¯ ¯ f (z)$ by the reflection principle, taking the integral path $C$ symmetric with respect to the real axis, that is, $φ (- u) = ¯ ¯¯¯¯¯¯¯¯¯ ¯ φ (u)$ , which leads to

φ^{'} (- u) = - ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ φ^{'} (u),

and taking the DE transform $ψ_{D E} (v)$ to be an even function, we have

	$I^{(n)} [f] ≃ I_{h, N}^{(n)'} [f]$
$\equiv$	$\frac{h}{2 π} Im {φ (ψ_{D E} (0))^{- n} f (φ (ψ_{D E} (0))) log (- φ (ψ_{D E} (0))) φ^{'} (ψ_{D E} (0)) ψ_{D E}^{'} (0)}$
	$+ \frac{h}{π} Im {N \sum k = 1 φ (ψ_{D E} (k h))^{- n} f (φ (ψ_{D E} (k h))) log (- φ (ψ_{D E} (k h)))$
	$\times φ^{'} (ψ_{D E} (k h)) ψ_{D E}^{'} (k h)} .$	(9)

3 Numerical examples

In this section, we show some numerical examples which show the effectiveness of the proposed method.

We computed the f.p. integrals

	$(i)$			(10)
	$(i i)$	$f . p . \int_{0}^{\infty} x^{- n} e^{- x} d x = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - γ & (n = 1) - 1 + γ & (n = 2) \frac{3}{4} - \frac{1}{2} γ & (n = 3) - \frac{11}{36} + \frac{1}{6} γ & (n = 4) \end{matrix}$		(10)

for $n = 1, 2, 3, 4$ , where $γ$ is Euler’s constant, by the formula (9). All the computations were performed using programs coded in C++ with double precision working. The complex integral path $C$ in (4) is taken as

C : z = φ (u) = \frac{u + 0.5 i}{i π} log (\frac{1 + i (u + 0.5 i)}{1 - i (u + 0.5 i)}), + \infty > u > - \infty

(see Figure 2). We took the number of sampling points $N$ for given mesh $h = 2^{- 1}, 2^{- 2}, \dots$ by truncating the infinite sum at the $k$ -th term such that

\frac{h}{π} \times | the k -th term | < {\begin{matrix} 10^{- 15} \times | I_{h, N}^{(n)'} [f] | & if I_{h, N}^{(n)'} [f] \neq 0 10^{- 15} & otherwise . \end{matrix}

Figure 2: The complex integral path $C$ .

Figure 3 shows the relative errors of the proposed approximation formula (9)

ε_{N}^{(n)} [f] = ⎧ ⎨ ⎩ \begin{matrix} | I_{h, N}^{(n)} [f] - I^{(n)} [f] | / | I^{(n)} [f] | & (I^{(n)} [f] \neq 0) | I_{h, N}^{(n)} [f] | & (otherwise) \end{matrix}

applied to the f.p. integrals (10). These figures shows the exponential convergence of the proposed formula as the number of sampling points $N$ increases.

Figure 3: The errors of the proposed approximation formula (9) applied to the f.p. integrals (10).

N $N$ e $ε_{N}^{(n)} [f]$	N $N$ e $ε_{N}^{(n)} [f]$
integral (i)	integral (ii)

4 Summary

In this paper, we proposed a numerical integration formula for Hadamard finite-part integrals with an integral power singularity at the endpoint on a half-infinite interval. In the proposed method, we express the desired f.p. integral using 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 the exponential convergence of the proposed method in the case that the integrand is an analytic function.

References

[1] B. Bialecki (1990) A sinc quadrature rule for hadamard finite-part integrals. Numer. Math. 57, pp. 263–269. External Links: Document Cited by: §1.
[2] B. Bialecki (1990) A sinc-hunter quadrature rule for cauchy principal value integrals. Math. Comput. 55, pp. 665–681. External Links: Document Cited by: §1.
[3] D. Elliot and D. F. Paget (1979) Gauss type quadrature rules for cauchy principal value integrals. Math. Comput. 33, pp. 301–309. External Links: Document Cited by: §1.
[4] R. Estrada and R. P. Kanwal (1989) Regularization, pseudofunction, and hadamard finite part. J. Math. Anal. Appl. 141, pp. 195–207. External Links: Document Cited by: §1.
[5] H. Ogata and H. Hirayama (2018) Numerical integration based on hyperfunction theory. J. Comput. Appl. Math. 327, pp. 243–259. Cited by: §1.
[6] H. Ogata, M. Sugihara, and M. Mori (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.
[7] H. Ogata (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.
[8] H. Ogata (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.
[9] D. F. Paget (1981) The numerical evaluation of hadamard finite-part integrals. Numer. Math. 36, pp. 447–453. External Links: Document Cited by: §1.
[10] F. Stenger (1993) Numerical methods based on sinc and analytic functions. Springer-Verlag, New York. Cited by: §2.
[11] H. Takahasi and M. Mori (1978) Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci., Kyoto Univ. 339, pp. 721–741. Cited by: §1, §1, §2.

A numerical method of computing Hadamard finite-part integrals with an integral power singularity at the endpoint on a half infinite interval

Numerical method of computing Hadamard finite-part integrals with a non-integral power singularity at the endpoint over a half infinite interval

Numerical method for computing Hadamard finite-part integrals with a non-integral power singularity at an endpoint

Computing the matrix fractional power based on the double exponential formula

A boundary integral method for 3D nonuniform dielectric waveguide problems via the windowed Green function

The Complex-Scaled Half-Space Matching Method

Matroid-Based TSP Rounding for Half-Integral Solutions

Interval probability density functions constructed from a generalization of the Moore and Yang integral

1 Introduction

2 Hadamard finite-part integrals and a numerical method

Theorem 1

Proof of Theorem 1

Theorem 2

Proof of Theorem 2

3 Numerical examples

4 Summary

References

		$N (f, φ, ψ_{D E}, D_{d})$
	$\equiv$	$lim ϵ \to 0 \oint_{\partial D_{d} (ϵ)} \| φ (ψ_{D E} (w))^{- n} f (φ (ψ_{D E} (w))) log (- φ (ψ_{D E} (w))) ψ_{D E}^{'} (w) \|$
	$<$	$\infty,$

A numerical method of computing Hadamard finite-part integrals with an integral power singularity at the endpoint on a half infinite interval

Related Research

Numerical method of computing Hadamard finite-part integrals with a non-integral power singularity at the endpoint over a half infinite interval

Numerical method for computing Hadamard finite-part integrals with a non-integral power singularity at an endpoint

Computing the matrix fractional power based on the double exponential formula

A boundary integral method for 3D nonuniform dielectric waveguide problems via the windowed Green function

The Complex-Scaled Half-Space Matching Method

Matroid-Based TSP Rounding for Half-Integral Solutions

Interval probability density functions constructed from a generalization of the Moore and Yang integral

1 Introduction

2 Hadamard finite-part integrals and a numerical method

Theorem 1

Proof of Theorem 1

Theorem 2

Proof of Theorem 2

3 Numerical examples

4 Summary

References