# The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

Authors: Jin Li
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernelcot
(
(
x
-
s
)
/
2
) is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.


---

## Body

## 1. Introduction

Consider the Cauchy principal integral(1)
I
f
;
s
=
⨍
c
c
+
2
π
⁡
cot
⁡
x
-
s
2
f
x
d
x
=
g
s
,
s
∈
0,2
π
, where ⨍
c
c
+
2
π denotes a Cauchy principal value integral and s is the singular point.There are several different definitions which can be proved equally, such as the definition of subtraction of the singularity, regularity definition, and direct definition.In this paper we adopt the following one:(2)
⨍
c
c
+
2
π
⁡
cot
⁡
x
-
s
2
f
x
d
x
=
l
i
m
ɛ
→
0
∫
0
s
-
ɛ
cot
⁡
x
-
s
2
f
x
d
x
+
∫
s
+
ɛ
2
π
cot
⁡
x
-
s
2
f
x
d
x
.Cauchy principal value integral has recently attracted a lot of attention; see, for example, [1]. The main reason for this interest is probably due to the fact that integral equations with Cauchy principal value integrals have been shown to be an adequate tool [1] for the modeling of many physical situations, such as acoustics, fluid mechanics, elasticity, fracture mechanics, and electromagnetic scattering problems. Numerous work has been devoted in developing efficient quadrature formulas, such as the Gaussian method [2–8], the Newton-Cotes methods [9–13], spline methods [14, 15], and some other methods [16–22].It is the aim of this paper to investigate the superconvergence phenomenon of trapezoidal rule and, in particular, to derive error estimates. In this paper the density functionf
(
x
) is replaced by the approximation function f
L
(
x
) while the singular kernel c
o
t
x
-
s
/
2 is computed c
o
t
x
i
-
s
/
2 in each subinterval, where f
L
(
x
) is the trapezoidal rule. This method is different from the semidiscrete methods and the order of singularity kernel can be reduced somehow which was firstly presented by Linz in the paper to calculate the hypersingular integral on interval. He used the trapezoidal rule and Simpson rule to approximate the density function f
(
x
) and the convergence rate was O
(
h
k
), k
=
1,2, when the singular point is always located at the middle of certain subinterval.The superconvergence of composite Newton-Cotes rules for Hadamard finite-part integrals was studied in [23, 24], where the superconvergence rate and the superconvergence point were presented, respectively. In [25, 26] the classical composite midpoint rectangle rule and classical composite trapezoidal rectangle rule for the computation of Cauchy principal value integrals are discussed. When the singular point coincides with some a priori known point, the convergence rate of the midpoint rectangle rule is higher than the global one, the same as the Riemann integral, which is called superconvergence phenomenon.This paper focuses on the superconvergence of trapezoidal rule for Cauchy principle integrals in a circle. Based on the investigation of the pointwise superconvergence phenomenon, that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. We prove both theoretically and numerically that the composite trapezoidal rule reaches the superconvergence rateO
(
h
2
) when the local coordinate of the singular point s is 0.The rest of this paper is organized as follows. In Section2, after introducing some basic formulas of the rectangle rule, we present the main results. In Section 3 we perform the proof. Finally, several numerical examples are provided to validate our analysis.


## 2. Main Result

Letc
=
x
0
<
x
1
<
⋯
<
x
n
-
1
<
x
n
=
c
+
2
π be a uniform partition of the interval [
c
,
c
+
2
π
] with mesh size h
=
2
π
/
n. Define by f
L
(
x
) the piecewise line interpolant for f
(
x
):(3)
f
L
x
=
f
x
i
2
,
i
=
0,1
,
…
,
n
,and a linear transformation(4)
x
=
x
^
i
τ
≔
τ
+
1
x
i
+
1
-
x
i
2
+
x
i
,
τ
∈
-
1,1
,from the reference element [
-
1,1
] to the subinterval [
x
i
,
x
i
+
1
]. Replacing f
(
x
) in (2) with f
L
(
x
) gives the composite trapezoidal rule:(5)
I
n
f
;
s
≔
∑
i
=
0
n
ω
i
s
f
x
i
=
I
f
;
s
-
E
n
f
,where ω
i
(
s
) denote the Cotes coefficients given by(6)
ω
i
s
=
h
2
cot
⁡
x
i
-
s
2
.We also define(7)
k
s
x
=
x
-
s
cot
⁡
x
-
s
2
,
x
≠
s
,
2
,
x
=
s
.Theorem 1.
Assumef
(
x
)
∈
C
1
[
a
,
b
]. For the trapezoidal rule I
n
(
f
,
s
) is defined as (5). Assuming that s
=
x
m
+
(
1
+
τ
)
h
/
2, there exists a positive constant C, independent of h and s, such that(8)
E
n
f
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
,where(9)
γ
τ
=
m
i
n
0
≤
i
≤
n
s
-
x
i
h
=
1
-
τ
2
.Proof.
LetR
(
x
)
=
f
(
x
)
-
f
(
x
i
)
/
2
-
f
(
x
i
+
1
)
/
2; then we have R
x
≤
C
h. Thus(10)
E
n
f
=
∑
i
=
0
n
⨍
x
i
x
i
+
1
⁡
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
=
∑
i
=
0
n
⨍
x
i
x
i
+
1
⁡
R
x
x
-
s
d
x
+
∑
i
=
0
n
⨍
x
i
x
i
+
1
⁡
k
s
x
-
2
x
-
s
R
x
d
x
+
∑
i
=
0
n
⨍
x
i
x
i
+
1
⁡
cot
⁡
x
-
s
2
-
cot
⁡
x
i
-
s
2
f
x
i
2
d
x
+
∑
i
=
0
n
⨍
x
i
x
i
+
1
⁡
cot
⁡
x
-
s
2
cot
⁡
x
i
+
1
-
s
2
f
x
i
+
1
2
d
x
.For the first part of (10), we have(11)
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
R
x
x
-
s
d
x
≤
C
max
⁡
k
s
x
h
∫
c
x
m
1
s
-
x
d
x
+
∫
x
m
+
1
c
+
2
π
1
x
-
s
d
x
=
C
max
⁡
k
s
x
h
ln
⁡
c
+
2
π
-
s
s
-
c
x
m
+
1
-
s
s
-
x
m
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
.For the second part of (10), we have(12)
⨍
x
m
x
m
+
1
⁡
R
x
x
-
s
d
x
≤
∫
x
m
x
m
+
1
R
x
-
R
s
x
-
s
d
x
+
R
s
ln
⁡
x
m
+
1
-
s
s
-
x
m
≤
C
h
ln
⁡
γ
τ
,
(13)
⨍
c
c
+
2
π
⁡
k
s
x
-
2
x
-
s
R
x
d
x
≤
C
h
⨍
c
c
+
2
π
⁡
k
s
x
-
2
x
-
s
d
x
≤
C
h
ln
⁡
γ
τ
.For the third part of (10), we have(14)
∑
i
=
0
n
⨍
x
i
x
i
+
1
⁡
cot
⁡
x
-
s
2
-
cot
⁡
x
i
-
s
2
f
x
i
2
d
x
≤
∑
i
=
0
,
i
≠
m
n
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
cot
⁡
x
i
-
s
2
f
x
i
2
d
x
+
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
-
cot
⁡
x
i
-
s
2
f
x
i
2
d
x
=
C
max
⁡
k
s
x
h
ln
⁡
c
+
2
π
-
s
s
-
c
x
m
+
1
-
s
s
-
x
m
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
.Combining (11), (13), and (14) together, the proof is completed.Set(15)
I
n
,
i
s
=
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
,
i
=
m
,
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
,
i
≠
m
.Lemma 2.
Assumes
=
x
m
+
(
τ
+
1
)
h
/
2 with τ
∈
(
-
1,1
). Let I
n
,
i
(
s
) be defined by (15); then there holds that(16)
I
n
,
i
s
=
-
2
∑
k
=
1
∞
1
k
cos
⁡
k
x
i
+
1
-
s
-
cos
⁡
k
x
i
-
s
+
h
∑
k
=
1
∞
sin
⁡
k
x
i
+
1
-
s
+
sin
⁡
k
x
i
-
s
.Proof.
Fori
=
m, by the definition of Cauchy principal value integral, we have(17)
I
n
,
m
s
=
lim
ɛ
→
0
⁡
∫
x
m
s
-
ɛ
+
∫
s
+
ɛ
x
m
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
m
-
s
2
-
1
2
cot
⁡
x
m
+
1
-
s
2
d
x
=
2
ln
⁡
2
sin
⁡
x
m
-
s
2
-
2
ln
⁡
2
sin
⁡
x
m
+
1
-
s
2
+
h
-
1
2
cot
⁡
x
m
-
s
2
-
1
2
cot
⁡
x
m
+
1
-
s
2
.For i
≠
m, taking integration by parts on the correspondent Riemann integral, we have(18)
I
n
,
i
s
=
2
ln
⁡
2
sin
⁡
x
i
-
s
2
-
2
ln
⁡
2
sin
⁡
x
i
+
1
-
s
2
+
h
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
.Now, by using the well-known identity,(19)
ln
⁡
2
sin
⁡
x
2
=
-
∑
n
=
1
∞
1
n
cos
⁡
n
x
,
1
2
cot
⁡
x
2
=
∑
n
=
1
∞
sin
⁡
n
x
.The proof is completed.Lemma 3.
Under the same assumptions of Lemma2, there holds that(20)
∑
i
=
1
n
I
n
,
i
s
=
-
2
π
tan
⁡
τ
π
2
.Proof.
By (15), we have(21)
∑
i
=
1
n
I
n
,
i
s
=
-
2
∑
k
=
1
∞
1
k
cos
⁡
k
x
i
+
1
-
s
-
cos
⁡
k
x
i
-
s
+
h
∑
k
=
1
∞
sin
⁡
k
x
i
+
1
-
s
+
sin
⁡
k
x
i
-
s
=
2
h
∑
k
=
1
∞
∑
i
=
1
n
sin
⁡
k
x
i
-
s
=
2
h
∑
j
=
1
∞
n
sin
⁡
n
j
x
1
-
s
=
4
π
∑
j
=
1
∞
sin
⁡
j
1
+
τ
π
=
2
π
cot
⁡
1
+
τ
π
2
=
-
2
π
tan
⁡
τ
π
2
.The proof is completed.Theorem 4.
Assumef
(
x
)
∈
C
2
[
a
,
b
]. For the trapezoidal rule I
n
(
f
,
s
) is defined as (5). Assuming that s
=
x
m
+
(
1
+
τ
)
h
/
2, there exists a positive constant C, independent of h and s, such that(22)
E
n
f
=
-
2
f
s
π
tan
⁡
τ
π
2
+
R
n
s
,where(23)
R
n
s
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
2
.
γ
(
τ
) is defined as (9).It is known that the global convergence rate of the composite trapezoidal rule is lower than Riemann integral.


## 3. Proof of the Theorem

In this section, we study the superconvergence of the composite trapezoidal rule for Cauchy principle integrals.


### 3.1. Preliminaries

In the following analysis,C will denote a generic constant that is independent of h and s and it may have different values in different places.Lemma 5.
Under the same assumptions of Theorem4, it holds that (24)
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
=
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
f
s
+
x
-
s
cot
⁡
x
-
s
2
-
x
i
-
s
2
cot
⁡
x
i
-
s
2
-
x
i
+
1
-
s
2
cot
⁡
x
i
+
1
-
s
2
f
′
s
+
R
f
1
x
+
R
f
2
x
+
R
f
3
x
+
R
f
4
x
,where(25)
R
f
1
x
=
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
,
R
f
2
x
=
-
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
,
R
f
3
x
=
-
f
′
′
α
i
2
2
x
i
+
1
-
x
2
cot
⁡
x
i
+
1
-
s
2
,
R
f
4
x
=
-
1
2
x
i
-
x
cot
⁡
x
i
-
s
2
+
x
i
+
1
-
x
cot
⁡
x
i
+
1
-
s
2
f
′
′
β
i
2
x
-
s
.Proof.
Performing Taylor expansion off
L
(
x
) at the point x, we have(26)
f
x
i
=
f
x
+
f
′
x
x
i
-
x
+
f
′
′
α
i
1
2
x
i
-
x
2
,
f
x
i
+
1
=
f
x
+
f
′
x
x
i
+
1
-
x
+
f
′
′
α
i
2
2
x
i
+
1
-
x
2
.Similarly, we have(27)
f
x
=
f
s
+
f
′
s
x
-
s
+
1
2
f
′
′
β
i
1
x
-
s
2
,
f
′
x
=
f
′
s
+
f
′
′
β
i
2
x
-
s
.We set(28)
E
m
x
=
cot
⁡
x
-
s
2
f
x
-
f
x
m
2
cot
⁡
x
m
-
s
2
-
f
x
m
+
1
2
cot
⁡
x
m
+
1
-
s
2
-
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
m
-
s
2
-
1
2
cot
⁡
x
m
+
1
-
s
2
f
s
+
x
-
s
cot
⁡
x
-
s
2
-
x
m
-
s
2
cot
⁡
x
m
-
s
2
-
x
m
+
1
-
s
2
cot
⁡
x
m
+
1
-
s
2
f
′
s
.Lemma 6.
Letf
(
x
)
∈
C
2
[
a
,
b
], denote E
m
(
x
) to be the error functional for the composite trapezoidal rule, and assume s
≠
x
i for any i
=
0,1
,
2
,
…
,
n; then there holds(29)
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
E
m
x
d
x
≤
C
h
2
ln
⁡
γ
τ
.Proof.
Asf
(
x
)
∈
C
2
[
a
,
b
], we get E
i
(
x
)
∈
C
2
[
a
,
b
]. We have(30)
⨍
x
m
x
m
+
1
⁡
c
o
t
x
-
s
2
E
m
x
d
x
=
2
⨍
x
m
x
m
+
1
⁡
E
m
x
x
-
s
d
x
+
⨍
x
m
x
m
+
1
⁡
k
s
x
-
2
x
-
s
E
m
x
d
x
.For the first part of (30), by the definition of(31)
⨍
a
b
⁡
f
x
x
-
s
d
x
=
∫
a
b
f
x
-
f
s
x
-
s
d
x
+
f
s
ln
⁡
b
-
s
s
-
a
,then we have(32)
⨍
x
m
x
m
+
1
⁡
E
m
x
x
-
s
d
x
=
∫
x
m
x
m
+
1
E
m
x
-
E
m
s
x
-
s
d
x
+
E
m
s
ln
⁡
x
m
+
1
-
s
s
-
x
m
;then we get(33)
⨍
x
m
x
m
+
1
⁡
E
m
x
x
-
s
d
x
≤
∫
x
m
x
m
+
1
E
m
x
-
E
m
s
x
-
s
d
x
+
E
m
s
ln
⁡
x
m
+
1
-
s
s
-
x
m
≤
C
h
2
ln
⁡
γ
τ
.For the second part of (30), we have(34)
⨍
x
m
x
m
+
1
⁡
k
s
x
-
2
x
-
s
R
x
d
x
≤
C
h
2
⨍
x
m
x
m
+
1
⁡
k
s
x
-
2
x
-
s
d
x
=
C
h
2
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
d
x
-
⨍
x
m
x
m
+
1
⁡
2
x
-
s
d
x
≤
C
h
2
ln
⁡
γ
τ
.And the proof is completed.Proof of Theorem4.
By Lemma5, we have(35)
∫
c
x
m
+
∫
x
m
+
1
c
+
2
π
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
=
f
s
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
f
x
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
f
′
s
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
x
-
s
cot
⁡
x
-
s
2
-
x
i
-
s
2
cot
⁡
x
i
-
s
2
-
x
i
+
1
-
s
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
2
2
x
i
+
1
-
x
2
cot
⁡
x
i
+
1
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
β
i
2
x
-
s
2
x
i
-
x
cot
⁡
x
i
-
s
2
+
x
i
+
1
-
x
cot
⁡
x
i
+
1
-
s
2
d
x
.By the definition of E
m
x, we have(36)
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
f
x
-
f
x
m
2
cot
⁡
x
m
-
s
2
-
f
x
m
+
1
2
cot
⁡
x
m
+
1
-
s
2
d
x
=
⨍
x
m
x
m
+
1
⁡
E
m
x
cot
⁡
x
-
s
2
d
x
+
f
s
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
m
-
s
2
-
1
2
cot
⁡
x
m
+
1
-
s
2
d
x
+
f
′
s
⨍
x
m
x
m
+
1
⁡
x
-
s
cot
⁡
x
-
s
2
-
x
m
-
s
2
cot
⁡
x
m
-
s
2
-
x
m
+
1
-
s
2
cot
⁡
x
m
+
1
-
s
2
d
x
.
Putting (35), (36) together yields(37)
⨍
c
c
+
2
π
⁡
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
f
x
-
f
x
m
2
cot
⁡
x
m
-
s
2
-
f
x
m
+
1
2
cot
⁡
x
m
+
1
-
s
2
d
x
=
f
s
∑
i
=
0
n
-
1
⨍
x
i
x
i
+
1
⁡
cot
⁡
x
-
s
2
f
x
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
f
′
s
∑
i
=
0
n
-
1
⨍
x
i
x
i
+
1
⁡
x
-
s
cot
⁡
x
-
s
2
-
x
i
-
s
2
cot
⁡
x
i
-
s
2
-
x
i
+
1
-
s
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
R
n
s
=
-
2
f
s
π
tan
⁡
τ
π
2
+
R
n
s
,where(38)
R
n
s
=
R
1
+
R
2
,
R
1
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
2
2
x
i
+
1
-
x
2
cot
⁡
x
i
+
1
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
β
i
2
x
-
s
2
x
i
-
x
cot
⁡
x
i
-
s
2
+
x
i
+
1
-
x
cot
⁡
x
i
+
1
-
s
2
d
x
,
R
2
=
⨍
x
m
x
m
+
1
⁡
E
m
x
c
o
t
x
-
s
2
d
x
.Now we estimate R
n
(
s
) term by term. For the first part of R
1, we have(39)
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
k
s
x
cot
⁡
x
-
s
/
2
-
1
/
2
cot
⁡
x
i
-
s
/
2
-
1
/
2
cot
⁡
x
i
+
1
-
s
/
2
1
/
2
f
′
′
β
i
1
x
-
s
3
x
-
s
d
x
≤
C
max
⁡
k
s
x
h
2
∫
c
x
m
1
s
-
x
d
x
+
∫
x
m
+
1
c
+
2
π
1
x
-
s
d
x
=
C
max
⁡
k
s
x
h
2
ln
⁡
c
+
2
π
-
s
s
-
c
x
m
+
1
-
s
s
-
x
m
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
2
.For the second part of R
1, there is no singularity and we have(40)
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
k
s
x
f
′
′
α
i
1
/
2
x
i
-
x
2
cot
⁡
x
i
-
s
/
2
x
-
s
d
x
≤
C
max
⁡
k
s
x
h
2
∫
c
x
m
1
s
-
x
d
x
+
∫
x
m
+
1
c
+
2
π
1
x
-
s
d
x
=
C
max
⁡
k
s
x
h
2
ln
⁡
c
+
2
π
-
s
s
-
c
x
m
+
1
-
s
s
-
x
m
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
2
.The third and fourth parts of R
1 can be similarly obtained. For R
2, by Lemmas 5 and 6, then we get(41)
R
n
s
≤
R
1
+
R
2
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
2and the proof is completed.Remark 7.
By the identity in [27],(42)
π
cot
⁡
π
x
=
∑
l
=
-
∞
l
=
∞
1
x
+
l
;then we get(43)
cot
⁡
x
-
s
2
=
2
x
-
s
+
∑
l
=
1
∞
2
x
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
-
s
+
2
l
π
,
cot
⁡
x
-
s
2
-
cot
⁡
x
^
m
-
s
2
=
2
x
-
s
+
∑
l
=
1
∞
2
x
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
-
s
+
2
l
π
-
2
x
^
m
-
s
+
∑
l
=
1
∞
2
x
^
m
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
^
m
-
s
+
2
l
π
=
2
x
^
m
-
x
x
-
s
x
^
m
-
s
+
∑
l
=
1
∞
2
x
^
m
-
x
x
-
s
-
2
l
π
x
^
m
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
^
m
-
x
x
-
s
+
2
l
π
x
^
m
-
s
+
2
l
π
.We can prove that the error expansion of this paper has the same error expansion of [25] for the Cauchy principal integral defined on the interval. Based on the error expansion, we get the same superconvergence point with local coordinate point equal to zero.
Based on Theorem4, we present the modified trapezoidal rule(44)
I
~
n
f
;
s
=
I
n
f
;
s
-
2
f
s
π
tan
⁡
τ
π
2
.


## 3.1. Preliminaries

In the following analysis,C will denote a generic constant that is independent of h and s and it may have different values in different places.Lemma 5.
Under the same assumptions of Theorem4, it holds that (24)
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
=
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
f
s
+
x
-
s
cot
⁡
x
-
s
2
-
x
i
-
s
2
cot
⁡
x
i
-
s
2
-
x
i
+
1
-
s
2
cot
⁡
x
i
+
1
-
s
2
f
′
s
+
R
f
1
x
+
R
f
2
x
+
R
f
3
x
+
R
f
4
x
,where(25)
R
f
1
x
=
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
,
R
f
2
x
=
-
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
,
R
f
3
x
=
-
f
′
′
α
i
2
2
x
i
+
1
-
x
2
cot
⁡
x
i
+
1
-
s
2
,
R
f
4
x
=
-
1
2
x
i
-
x
cot
⁡
x
i
-
s
2
+
x
i
+
1
-
x
cot
⁡
x
i
+
1
-
s
2
f
′
′
β
i
2
x
-
s
.Proof.
Performing Taylor expansion off
L
(
x
) at the point x, we have(26)
f
x
i
=
f
x
+
f
′
x
x
i
-
x
+
f
′
′
α
i
1
2
x
i
-
x
2
,
f
x
i
+
1
=
f
x
+
f
′
x
x
i
+
1
-
x
+
f
′
′
α
i
2
2
x
i
+
1
-
x
2
.Similarly, we have(27)
f
x
=
f
s
+
f
′
s
x
-
s
+
1
2
f
′
′
β
i
1
x
-
s
2
,
f
′
x
=
f
′
s
+
f
′
′
β
i
2
x
-
s
.We set(28)
E
m
x
=
cot
⁡
x
-
s
2
f
x
-
f
x
m
2
cot
⁡
x
m
-
s
2
-
f
x
m
+
1
2
cot
⁡
x
m
+
1
-
s
2
-
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
m
-
s
2
-
1
2
cot
⁡
x
m
+
1
-
s
2
f
s
+
x
-
s
cot
⁡
x
-
s
2
-
x
m
-
s
2
cot
⁡
x
m
-
s
2
-
x
m
+
1
-
s
2
cot
⁡
x
m
+
1
-
s
2
f
′
s
.Lemma 6.
Letf
(
x
)
∈
C
2
[
a
,
b
], denote E
m
(
x
) to be the error functional for the composite trapezoidal rule, and assume s
≠
x
i for any i
=
0,1
,
2
,
…
,
n; then there holds(29)
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
E
m
x
d
x
≤
C
h
2
ln
⁡
γ
τ
.Proof.
Asf
(
x
)
∈
C
2
[
a
,
b
], we get E
i
(
x
)
∈
C
2
[
a
,
b
]. We have(30)
⨍
x
m
x
m
+
1
⁡
c
o
t
x
-
s
2
E
m
x
d
x
=
2
⨍
x
m
x
m
+
1
⁡
E
m
x
x
-
s
d
x
+
⨍
x
m
x
m
+
1
⁡
k
s
x
-
2
x
-
s
E
m
x
d
x
.For the first part of (30), by the definition of(31)
⨍
a
b
⁡
f
x
x
-
s
d
x
=
∫
a
b
f
x
-
f
s
x
-
s
d
x
+
f
s
ln
⁡
b
-
s
s
-
a
,then we have(32)
⨍
x
m
x
m
+
1
⁡
E
m
x
x
-
s
d
x
=
∫
x
m
x
m
+
1
E
m
x
-
E
m
s
x
-
s
d
x
+
E
m
s
ln
⁡
x
m
+
1
-
s
s
-
x
m
;then we get(33)
⨍
x
m
x
m
+
1
⁡
E
m
x
x
-
s
d
x
≤
∫
x
m
x
m
+
1
E
m
x
-
E
m
s
x
-
s
d
x
+
E
m
s
ln
⁡
x
m
+
1
-
s
s
-
x
m
≤
C
h
2
ln
⁡
γ
τ
.For the second part of (30), we have(34)
⨍
x
m
x
m
+
1
⁡
k
s
x
-
2
x
-
s
R
x
d
x
≤
C
h
2
⨍
x
m
x
m
+
1
⁡
k
s
x
-
2
x
-
s
d
x
=
C
h
2
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
d
x
-
⨍
x
m
x
m
+
1
⁡
2
x
-
s
d
x
≤
C
h
2
ln
⁡
γ
τ
.And the proof is completed.Proof of Theorem4.
By Lemma5, we have(35)
∫
c
x
m
+
∫
x
m
+
1
c
+
2
π
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
=
f
s
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
f
x
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
f
′
s
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
x
-
s
cot
⁡
x
-
s
2
-
x
i
-
s
2
cot
⁡
x
i
-
s
2
-
x
i
+
1
-
s
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
2
2
x
i
+
1
-
x
2
cot
⁡
x
i
+
1
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
β
i
2
x
-
s
2
x
i
-
x
cot
⁡
x
i
-
s
2
+
x
i
+
1
-
x
cot
⁡
x
i
+
1
-
s
2
d
x
.By the definition of E
m
x, we have(36)
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
f
x
-
f
x
m
2
cot
⁡
x
m
-
s
2
-
f
x
m
+
1
2
cot
⁡
x
m
+
1
-
s
2
d
x
=
⨍
x
m
x
m
+
1
⁡
E
m
x
cot
⁡
x
-
s
2
d
x
+
f
s
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
m
-
s
2
-
1
2
cot
⁡
x
m
+
1
-
s
2
d
x
+
f
′
s
⨍
x
m
x
m
+
1
⁡
x
-
s
cot
⁡
x
-
s
2
-
x
m
-
s
2
cot
⁡
x
m
-
s
2
-
x
m
+
1
-
s
2
cot
⁡
x
m
+
1
-
s
2
d
x
.
Putting (35), (36) together yields(37)
⨍
c
c
+
2
π
⁡
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
f
x
-
f
x
i
2
cot
⁡
x
i
-
s
2
-
f
x
i
+
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
⨍
x
m
x
m
+
1
⁡
cot
⁡
x
-
s
2
f
x
-
f
x
m
2
cot
⁡
x
m
-
s
2
-
f
x
m
+
1
2
cot
⁡
x
m
+
1
-
s
2
d
x
=
f
s
∑
i
=
0
n
-
1
⨍
x
i
x
i
+
1
⁡
cot
⁡
x
-
s
2
f
x
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
f
′
s
∑
i
=
0
n
-
1
⨍
x
i
x
i
+
1
⁡
x
-
s
cot
⁡
x
-
s
2
-
x
i
-
s
2
cot
⁡
x
i
-
s
2
-
x
i
+
1
-
s
2
cot
⁡
x
i
+
1
-
s
2
d
x
+
R
n
s
=
-
2
f
s
π
tan
⁡
τ
π
2
+
R
n
s
,where(38)
R
n
s
=
R
1
+
R
2
,
R
1
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
2
2
x
i
+
1
-
x
2
cot
⁡
x
i
+
1
-
s
2
d
x
-
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
β
i
2
x
-
s
2
x
i
-
x
cot
⁡
x
i
-
s
2
+
x
i
+
1
-
x
cot
⁡
x
i
+
1
-
s
2
d
x
,
R
2
=
⨍
x
m
x
m
+
1
⁡
E
m
x
c
o
t
x
-
s
2
d
x
.Now we estimate R
n
(
s
) term by term. For the first part of R
1, we have(39)
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
cot
⁡
x
-
s
2
-
1
2
cot
⁡
x
i
-
s
2
-
1
2
cot
⁡
x
i
+
1
-
s
2
1
2
f
′
′
β
i
1
x
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
k
s
x
cot
⁡
x
-
s
/
2
-
1
/
2
cot
⁡
x
i
-
s
/
2
-
1
/
2
cot
⁡
x
i
+
1
-
s
/
2
1
/
2
f
′
′
β
i
1
x
-
s
3
x
-
s
d
x
≤
C
max
⁡
k
s
x
h
2
∫
c
x
m
1
s
-
x
d
x
+
∫
x
m
+
1
c
+
2
π
1
x
-
s
d
x
=
C
max
⁡
k
s
x
h
2
ln
⁡
c
+
2
π
-
s
s
-
c
x
m
+
1
-
s
s
-
x
m
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
2
.For the second part of R
1, there is no singularity and we have(40)
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
f
′
′
α
i
1
2
x
i
-
x
2
cot
⁡
x
i
-
s
2
d
x
=
∑
i
=
0
,
i
≠
m
n
-
1
∫
x
i
x
i
+
1
k
s
x
f
′
′
α
i
1
/
2
x
i
-
x
2
cot
⁡
x
i
-
s
/
2
x
-
s
d
x
≤
C
max
⁡
k
s
x
h
2
∫
c
x
m
1
s
-
x
d
x
+
∫
x
m
+
1
c
+
2
π
1
x
-
s
d
x
=
C
max
⁡
k
s
x
h
2
ln
⁡
c
+
2
π
-
s
s
-
c
x
m
+
1
-
s
s
-
x
m
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
2
.The third and fourth parts of R
1 can be similarly obtained. For R
2, by Lemmas 5 and 6, then we get(41)
R
n
s
≤
R
1
+
R
2
≤
C
max
⁡
k
s
x
ln
⁡
h
+
ln
⁡
γ
τ
h
2and the proof is completed.Remark 7.
By the identity in [27],(42)
π
cot
⁡
π
x
=
∑
l
=
-
∞
l
=
∞
1
x
+
l
;then we get(43)
cot
⁡
x
-
s
2
=
2
x
-
s
+
∑
l
=
1
∞
2
x
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
-
s
+
2
l
π
,
cot
⁡
x
-
s
2
-
cot
⁡
x
^
m
-
s
2
=
2
x
-
s
+
∑
l
=
1
∞
2
x
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
-
s
+
2
l
π
-
2
x
^
m
-
s
+
∑
l
=
1
∞
2
x
^
m
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
^
m
-
s
+
2
l
π
=
2
x
^
m
-
x
x
-
s
x
^
m
-
s
+
∑
l
=
1
∞
2
x
^
m
-
x
x
-
s
-
2
l
π
x
^
m
-
s
-
2
l
π
+
∑
l
=
1
∞
2
x
^
m
-
x
x
-
s
+
2
l
π
x
^
m
-
s
+
2
l
π
.We can prove that the error expansion of this paper has the same error expansion of [25] for the Cauchy principal integral defined on the interval. Based on the error expansion, we get the same superconvergence point with local coordinate point equal to zero.
Based on Theorem4, we present the modified trapezoidal rule(44)
I
~
n
f
;
s
=
I
n
f
;
s
-
2
f
s
π
tan
⁡
τ
π
2
.


## 4. Numerical Example

In this section, computational results are reported to confirm our theoretical analysis.Example. We consider the singular integral with f
(
x
)
=
2
cos
⁡
x
+
sin
⁡
x
c
=
0. s
=
x
[
n
/
4
]
+
(
τ
+
1
)
h
/
2 and s
=
x
0
+
(
τ
+
1
)
h
/
2 with τ
=
0 are the superconvergence point.From Tables1 and 2, we know that the superconvergence point is 0 with the coordinate location of singular point equal to zero, while, for the local coordinate of singular point not equal to zero, it is not convergence in general which coincides with our analysis.Table 1
An error estimate of the trapezoidal rule withs
=
x
[
n
/
4
]
+
(
τ
+
1
)
h
/
2.
τ
=
0
τ
=
-
2
3
τ
=
2
3
τ
=
1
2
8
0
-
9.3692
e
+
000
2.0089
e
+
000
1.7335
e
+
000
16
-
8.8818
e
-
016
-
1.0148
e
+
001
6.8071
e
+
000
4.1887
e
+
000
32
1.1546
e
-
014
-
1.0521
e
+
001
8.9647
e
+
000
5.2932
e
+
000
64
2.6645
e
-
015
-
1.0703
e
+
001
9.9570
e
+
000
5.8039
e
+
000
128
-
4.2633
e
-
014
-
1.0793
e
+
001
1.0429
e
+
001
6.0477
e
+
000
256
-
8.8818
e
-
015
-
1.0838
e
+
001
1.0658
e
+
001
6.1665
e
+
000Table 2
An error estimate of the trapezoidal rule withs
=
x
0
+
(
τ
+
1
)
h
/
2.
τ
=
0
τ
=
-
2
3
τ
=
2
3
τ
=
1
2
8
2.6645
e
-
015
-
1.2210
e
+
001
1.5259
e
+
001
8.7150
e
+
000
16
1.5099
e
-
014
-
1.1571
e
+
001
1.3803
e
+
001
7.8365
e
+
000
32
5.3291
e
-
015
-
1.1233
e
+
001
1.2510
e
+
001
7.1371
e
+
000
64
-
3.0198
e
-
014
-
1.1059
e
+
001
1.1736
e
+
001
6.7284
e
+
000
128
2.4869
e
-
014
-
1.0971
e
+
001
1.1319
e
+
001
6.5102
e
+
000
256
-
2.6645
e
-
015
-
1.0927
e
+
001
1.1103
e
+
001
6.3978
e
+
000For the modified classical trapezoidal rule, from Tables3 and 4, for the nonsuperconvergence point and the superconvergence point, we all get the superconvergence phenomenon.Table 3
An error estimate of the modified trapezoidal rule withs
=
x
[
n
/
4
]
+
(
τ
+
1
)
h
/
2.
τ
=
0
τ
=
-
2
3
τ
=
2
3
τ
=
1
2
8
0
3.5527
e
-
015
8.8818
e
-
016
0
16
-
8.8818
e
-
016
-
1.4211
e
-
014
1.1768
e
-
014
-
5.3291
e
-
015
32
1.1546
e
-
014
-
2.1316
e
-
014
1.5099
e
-
014
-
7.9936
e
-
015
64
2.6645
e
-
015
8.1712
e
-
014
-
5.5511
e
-
014
-
4.2744
e
-
014
128
-
4.2633
e
-
014
6.3949
e
-
014
-
6.1284
e
-
014
3.9413
e
-
014
256
-
8.8818
e
-
015
-
2.9132
e
-
013
2.4958
e
-
013
5.4956
e
-
015Table 4
An error estimate of the modified trapezoidal rule withs
=
x
0
+
(
τ
+
1
)
h
/
2.
τ
=
0
τ
=
-
2
3
τ
=
2
3
τ
=
1
2
8
2.6645
e
-
015
-
7.9936
e
-
015
2.1316
e
-
014
-
1.2434
e
-
014
16
1.5099
e
-
014
4.7962
e
-
014
3.1974
e
-
014
-
8.8818
e
-
015
32
5.3291
e
-
015
1.2079
e
-
013
-
6.7502
e
-
014
-
4.6185
e
-
014
64
-
3.0198
e
-
014
3.2863
e
-
013
-
7.4607
e
-
014
6.3949
e
-
014
128
2.4869
e
-
014
2.4780
e
-
013
2.4158
e
-
013
3.1974
e
-
014
256
-
2.6645
e
-
015
8.0824
e
-
013
-
4.5830
e
-
013
4.1744
e
-
013



---

*Source: 918083-2015-10-05.xml*