Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities
Authors: Sidi Avram 
Published: April 2012

In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the divergent finite-range integrals

∫
a
b

f
(
x
)

d
x

\int ^b_af(x)\,dx

, where

f
∈

C

∞

(
a
,
b
)

f\in C^{\infty }(a,b)

but can have arbitrary algebraic singularities at one or both endpoints. We assume that

f
(
x
)

f(x)

has asymptotic expansions of the general forms
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon f left-parenthesis x right-parenthesis tilde upper K left-parenthesis x minus a right-parenthesis Superscript negative 1 Baseline plus sigma-summation Underscript s equals 0 Overscript normal infinity Endscripts c Subscript s Baseline left-parenthesis x minus a right-parenthesis Superscript gamma Super Subscript s Superscript Baseline as x right-arrow a plus comma 2nd Row 1st Column Blank 2nd Column a m p semicolon f left-parenthesis x right-parenthesis tilde upper L left-parenthesis b minus x right-parenthesis Superscript negative 1 Baseline plus sigma-summation Underscript s equals 0 Overscript normal infinity Endscripts d Subscript s Baseline left-parenthesis b minus x right-parenthesis Superscript delta Super Subscript s Superscript Baseline as x right-arrow b minus comma EndLayout">

a
m
p
;
f
(
x
)
∼
K

(
x
−
a

)

−
1

+

∑

s
=
0

∞

c
s

(
x
−
a

)

γ
s

as
 
x
→
a
+
,

a
m
p
;
f
(
x
)
∼
L

(
b
−
x

)

−
1

+

∑

s
=
0

∞

d
s

(
b
−
x

)

δ
s

as
 
x
→
b
−
,

\begin{align*} &f(x)\sim K\,(x-a)^{-1}+\sum ^{\infty }_{s=0}c_s(x-a)^{\gamma _s} \quad \text {as}\ x\to a+,\\ &f(x)\sim L\,(b-x)^{-1}+\sum ^{\infty }_{s=0}d_s(b-x)^{\delta _s} \quad \text {as}\ x\to b-, \end{align*}

where

K
,
L

K,L

, and

c
s

,

d
s

c_s, d_s

,

s
=
0
,
1
,
…
,

s=0,1,\ldots ,

are some constants,

|

K

|

+

|

L

|

≠
0
,

|K|+|L|\neq 0,

and

γ
s

\gamma _s

and

δ
s

\delta _s

are distinct, arbitrary and, in general, complex, and different from

−
1

-1

, and satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal fraktur upper R gamma 0 less-than-or-equal-to normal fraktur upper R gamma 1 less-than-or-equal-to midline-horizontal-ellipsis comma limit Underscript s right-arrow normal infinity Endscripts normal fraktur upper R gamma Subscript s Baseline equals plus normal infinity semicolon normal fraktur upper R delta 0 less-than-or-equal-to normal fraktur upper R delta 1 less-than-or-equal-to midline-horizontal-ellipsis comma limit Underscript s right-arrow normal infinity Endscripts normal fraktur upper R delta Subscript s Baseline equals plus normal infinity period">

ℜ

γ
0

≤
ℜ

γ
1

≤
⋯
,
 
 

lim

s
→
∞

ℜ

γ
s

=
+
∞
;

ℜ

δ
0

≤
ℜ

δ
1

≤
⋯
,
 
 

lim

s
→
∞

ℜ

δ
s

=
+
∞
.

\Re \gamma _0\leq \Re \gamma _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \gamma _s=+\infty ;\quad \Re \delta _0\leq \Re \delta _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \delta _s=+\infty .

\]
Hence the integral

∫
a
b

f
(
x
)

d
x

\int ^b_af(x)\,dx

exists in the sense of Hadamard finite part. The results we obtain in this work extend some of the results in [A. Sidi, Numer. Math. 98 (2004), pp. 371–387] that pertain to the cases in which

K
=
L
=
0.

K=L=0.

They are expressed in very simple terms based only on the asymptotic expansions of

f
(
x
)

f(x)

as

x
→
a
+

x\to a+

and

x
→
b
−

x\to b-

. With

h
=
(
b
−
a
)

/

n

h=(b-a)/n

, where

n
n

is a positive integer, one of these results reads
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column h sigma-summation Underscript i equals 1 Overscript n minus 1 Endscripts f left-parenthesis a plus i h right-parenthesis tilde upper I left-bracket f right-bracket 2nd Column a m p semicolon plus upper K left-parenthesis upper C minus log h right-parenthesis plus sigma-summation Underscript StartLayout 1st Row s equals 0 2nd Row gamma Subscript s Baseline not-an-element-of StartSet 2 comma 4 comma ellipsis EndSet EndLayout Overscript normal infinity Endscripts c Subscript s Baseline zeta left-parenthesis minus gamma Subscript s Baseline right-parenthesis h Superscript gamma Super Subscript s Superscript plus 1 Baseline 2nd Row 1st Column Blank 2nd Column a m p semicolon plus upper L left-parenthesis upper C minus log h right-parenthesis plus sigma-summation Underscript StartLayout 1st Row s equals 0 2nd Row delta Subscript s Baseline not-an-element-of StartSet 2 comma 4 comma ellipsis EndSet EndLayout Overscript normal infinity Endscripts d Subscript s Baseline zeta left-parenthesis minus delta Subscript s Baseline right-parenthesis h Superscript delta Super Subscript s Superscript plus 1 Baseline as h right-arrow 0 comma EndLayout">

h

∑

i
=
1

n
−
1

f
(
a
+
i
h
)
∼
I
[
f
]

a
m
p
;
+
K

(
C
−
log
⁡
h
)
+

∑

s
=
0

γ
s

∉
{
2
,
4
,
…
}

∞

c
s

ζ
(
−

γ
s

)

h

γ
s

+
1

a
m
p
;
+
L

(
C
−
log
⁡
h
)
+

∑

s
=
0

δ
s

∉
{
2
,
4
,
…
}

∞

d
s

ζ
(
−

δ
s

)

h

δ
s

+
1

as 

h
→
0

,

\begin{align*} h\sum ^{n-1}_{i=1}f(a+ih)\sim I[f]&+K\,(C -\log h) + \sum ^{\infty }_{\substack {s=0\\ \gamma _s\not \in \{2,4,\ldots \}}}c_s \zeta (-\gamma _s)\,h^{\gamma _s+1}\\ &+L\,(C -\log h) +\sum ^{\infty }_{\substack {s=0\\ \delta _s\not \in \{2,4,\ldots \}}}d_s\zeta (-\delta _s) h^{\delta _s+1}\quad \text {as $h\to 0$}, \end{align*}

where

I
[
f
]

I[f]

is the Hadamard finite part of

∫
a
b

f
(
x
)

d
x

\int ^b_af(x)\,dx

,

C
C

is Euler’s constant and

ζ
(
z
)

\zeta (z)

is the Riemann Zeta function. We illustrate the results with an example.