International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 2 (2016), 101-109
http://www.etamaths.com

SOME DISCUSSIONS ON A KIND OF IMPROPER INTEGRALS

FENG QI1,2,3,∗ AND VIERA ČERŇANOVÁ4

Abstract. In the paper, the improper integral

I(a,b; λ,η) =

∫ b
a

1√
(t−a)(b− t)

lnλ t

tη
d t

for b > a > 0 and λ,η ∈ R is discussed, some explicit formulas for special cases of I(a,b; λ,η) are
presented, and several identities of I(a,b; k,η) for k ∈ N are established.

1. Motivation

The motivation of this paper origins from investigating central Delanoy numbers in [11]. For proving
the main result [11, Theorem 1.4], we need [11, Lemmas 2.4 and 2.5]. Lemma 2.4 in [11] states that,
for b > a and z ∈ C \ (−∞,−a], the principal branch of the function 1√

(z+a)(z+b)
can be represented

as
1√

(z + a)(z + b)
=

1

π

∫ b
a

1√
(t−a)(b− t)

1

t + z
d t, (1.1)

where C denotes the complex plane. When taking z = 0, the integral representation (1.1) becomes∫ b
a

1√
(t−a)(b− t)

1

t
d t =

π
√
ab
, b > a > 0. (1.2)

Lemma 2.5 in [11] reads that the improper integral

∫ α
1/α

1√
(t− 1/α)(α− t)

ln2k−1 t

tβ
d t



< 0, β > 1

2

= 0, β = 1
2

> 0, β < 1
2

(1.3)

for all k ∈ N, where α > 1 and β ∈ R.
Motivated by the above results, we naturally introduce the improper integral

I(a,b; λ,η) =

∫ b
a

1√
(t−a)(b− t)

lnλ t

tη
d t

=

∫ 1
0

1√
s(1 −s)

lnλ[(b−a)s + a]
[(b−a)s + a]η

d s

for b > a > 0 and λ,η ∈ R and consider a problem: how to compute the improper integral I(a,b; λ,η)?

2. Explicit formulas for special cases of I(a,b; λ,η)

In this section, we present several explicit formulas for special cases of the improper integral
I(a,b; λ,η).

In the monograph [4], we do not find such a kind of integrals I(a,b; λ,η) for general b > a > 0 and
λ,η ∈ R.

2010 Mathematics Subject Classification. Primary 40C10; Secondary 26A39, 26A42, 33C05, 33C20, 33E05, 33E20,
40A10, 97I50.

Key words and phrases. improper integral; explicit formula; identity; hypergeometric function; generalized hyperge-

ometric series.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

101



102 QI AND ČERŇANOVÁ

2.1. From (1.1) or (1.2), it follows that

I(a,b; 0, 1) =
π
√
ab
, b > a > 0. (2.1)

2.2. From (1.3), it follows that

I

(
a,

1

a
; 2k − 1,

1

2

)
= 0, 0 < a < 1, k ∈ N.

2.3. It is straightforward by using Euler’s substitution that

I(a,b; 0, 0) =

∫ 1
0

1√
s(1 −s)

d s = π, b > a > 0.

2.4. When λ = 0, η 6= 0, and 2a > b > a > 0, we have

I(a,b; 0,η) =
1

aη

∫ 1
0

[1 + (b/a− 1)s]−η√
s(1 −s)

d s

=
1

aη

∫ 1
0

(1 −s)−1/2
∞∑
`=0

〈−η〉`
(
b

a
− 1
)`
s`−1/2

`!
d s

=
1

aη

∞∑
`=0

〈−η〉`
`!

(
b

a
− 1
)` ∫ 1

0

(1 −s)−1/2s`−1/2 d s

=
1

aη

∞∑
`=0

〈−η〉`
`!

(
b

a
− 1
)`
B

(
1

2
,` +

1

2

)

=
1

aη

∞∑
`=0

(η)`
Γ(1/2)Γ(` + 1/2)

Γ(` + 1)

1

`!

(
1 −

b

a

)`
=

π

aη

∞∑
`=0

(η)`(1/2)`
(1)`

1

`!

(
1 −

b

a

)`
=

π

aη
2F1

(
η,

1

2
; 1; 1 −

b

a

)
,

where

〈x〉n =
n−1∏
k=0

(x−k) =

{
x(x− 1) · · ·(x−n + 1), n ≥ 1
1, n = 0

and

(x)` =

`−1∏
k=0

(x + k) =

{
x(x + 1)(x + 2) · · ·(x + `− 1), ` ≥ 1
1, ` = 0

are respectively called the falling and rising factorials of x ∈ R, the function B(x,y) denotes the
classical beta function, and 2F1 are the classical hypergeometric functions which are special cases of
the generalized hypergeometric series

pFq(a1, . . . ,ap; b1, . . . ,bq; z) =

∞∑
n=0

(a1)n . . . (ap)n
(b1)n . . . (bq)n

zn

n!

for complex numbers ai ∈ C and bi ∈ C\{0,−1,−2, . . .} and for positive integers p,q ∈ N. This result

I(a,b; 0,η) =
π

aη
2F1

(
η,

1

2
; 1; 1 −

b

a

)
, η 6= 0, 2a > b > a > 0

can also be found in [3, p. xv, eq. (12)].



SOME DISCUSSIONS ON A KIND OF IMPROPER INTEGRALS 103

2.5. When λ = k ∈ N and 2a > b > a > 0, the function lnk[(b−a)s + a] can be rewritten as

lnk[(b−a)s + a] =
(

ln a + ln

[
1 +

(
b

a
− 1
)
s

])k
=

k∑
`=0

(
k

`

)
lnk−` a ln`

[
1 +

(
b

a
− 1
)
s

]

=
(
lnk a

) k∑
`=0

(
k

`

)
(−1)`

ln` a

[
∞∑
m=1

1

m

(
1 −

b

a

)m
sm

]`

=
(
lnk a

) k∑
`=0

(
k

`

)
(−1)`

ln` a
s`

[
∞∑
m=0

1

m + 1

(
1 −

b

a

)m+1
sm

]`
.

When 0 < a < b < 1 or 1 < a < b < a2, if λ ∈ R, then

lnλ[(b−a)s + a] =
(
lnλ a

)(
1 +

ln[1 + (b/a− 1)s]
ln a

)λ
=
(
lnλ a

) ∞∑
`=0

〈λ〉`
`!

(
ln[1 + (b/a− 1)s]

ln a

)`

=
(
lnλ a

) ∞∑
`=0

(−1)`〈λ〉`
`! ln` a

[
∞∑
m=1

1

m

(
1 −

b

a

)m
sm

]`

=
(
lnλ a

) ∞∑
`=0

(−1)`〈λ〉`
`! ln` a

s`

[
∞∑
m=0

1

m + 1

(
1 −

b

a

)m+1
sm

]`
.

In [4, p. 18, 0.314], it was stated that(
∞∑
k=0

akx
k

)n
=

∞∑
k=0

cn,kx
k,

where cn,0 = a
n
0 and

cn,m =
1

ma0

m∑
k=1

(kn−m + k)akcn,m−k, m ∈ N.

Hence, it follows that [
∞∑
m=0

1

m + 1

(
1 −

b

a

)m+1
sm

]`
=

∞∑
m=0

c`,mx
m,

where c`,0 =
(
1 − b

a

)`
and

c`,m =
1

m

m∑
k=1

k`−m + k
k + 1

(
1 −

b

a

)k
c`,m−k

=
1

m

(
1 −

b

a

)m m−1∑
p=0

m`− (` + 1)p
m−p + 1

(
1 −

b

a

)−p
c`,p

for m ∈ N. Let C`,m =
(
1 − b

a

)−m
c`,m, the above recursive formula becomes

C`,m =
1

m

m−1∑
p=0

m`−p(` + 1)
m−p + 1

C`,p (2.2)



104 QI AND ČERŇANOVÁ

with C`,0 = c`,0. Starting out from these points, it is much possible to find explicit formulas for
computing the integral I(a,b; λ,η). For example, when λ 6= 0 and η = 1,

I(a,b; λ, 1) =
1

(λ + 1)(b−a)

∫ 1
0

1√
s(1 −s)

d lnλ+1[(b−a)s + a]
d s

d s

=
lnλ+1 a

(λ + 1)(b−a)

∞∑
`=0

(−1)`〈λ + 1〉`
`! ln` a

×
∫ 1

0

1√
s(1 −s)

d

d s

[
∞∑
m=0

1

m

(
1 −

b

a

)m
sm

]`
d s

=
lnλ+1 a

(λ + 1)(b−a)

∞∑
`=0

(−1)`〈λ + 1〉`
`! ln` a

∫ 1
0

1√
s(1 −s)

d

d s

∞∑
m=0

c`,ms
m d s

=
lnλ+1 a

(λ + 1)(b−a)

∞∑
`=0

(−1)`〈λ + 1〉`
`! ln` a

∞∑
m=0

(m + 1)c`,m+1

∫ 1
0

(1 −s)−1/2sm−1/2 d s

=
lnλ+1 a

(λ + 1)(b−a)

∞∑
`=0

(−1)`〈λ + 1〉`
`! ln` a

∞∑
m=0

(m + 1)c`,m+1B

(
1

2
,m +

1

2

)

=
π lnλ+1 a

(λ + 1)(b−a)

∞∑
`=0

(−1)`〈λ + 1〉`
`! ln` a

∞∑
m=0

(m + 1)c`,m+1
(1/2)m
(1)m

.

Hence, it would be important to derive a general formula for the recursive relation (2.2).

2.6. For k ≥ 0, differentiating with respect to z on both sides of (1.1) gives

dk

d zk
1√

(z + a)(z + b)
= (−1)k

k!

π

∫ b
a

1√
(t−a)(b− t)

1

(t + z)k+1
d t. (2.3)

By the Faá di Bruno formula

dn

d tn
f ◦h(t) =

n∑
k=0

f(k)(h(t))Bn,k
(
h′(t),h′′(t), . . . ,h(n−k+1)(t)

)
, n ≥ 0

in [2, p. 139, Theorem C], where

Bn,k(x1,x2, . . . ,xn−k+1) =
∑

1≤i≤n,`i∈N∪{0}∑n
i=1 i`i=n∑n
i=1

`i=k

n!∏n−k+1
i=1 `i!

n−k+1∏
i=1

(xi
i!

)`i
, n ≥ k ≥ 0

is called [2, p. 134, Theorem A] the Bell polynomials of the second kind, we obtain

dk

d zk
1√

(z + a)(z + b)
=

k∑
`=0

(
1
√
u

)(`)
Bk,`(u

′(z),u′′(z), 0, . . . , 0)

=

k∑
`=0

〈
−

1

2

〉
`

1

u`+1/2
Bk,`(2z + a + b, 2, 0, . . . , 0)

=

k∑
`=0

〈
−

1

2

〉
`

1

[(z + a)(z + b)]`+1/2
Bk,`(2z + a + b, 2, 0, . . . , 0)

→
k∑
`=0

〈
−

1

2

〉
`

1

(ab)`+1/2
Bk,`(a + b, 2, 0, . . . , 0)

as z → 0, where u = u(z) = (z + a)(z + b). Recall from [2, p. 135] that

Bn,k
(
abx1,ab

2x2, . . . ,ab
n−k+1xn−k+1

)
= akbnBn,k(x1,x2, . . . ,xn−k+1), (2.4)



SOME DISCUSSIONS ON A KIND OF IMPROPER INTEGRALS 105

where a and b are any complex numbers and n ≥ k ≥ 0. Recall from [5, Theoem 4.1], [17, Theorem 3.1],
and [18, Lemma 2.5] that

Bn,k(x, 1, 0, . . . , 0) =
(n−k)!

2n−k

(
n

k

)(
k

n−k

)
x2k−n, n ≥ k ≥ 0. (2.5)

Accordingly, by (2.4) and (2.5), it follows that

lim
z→0

dk

d zk
1√

(z + a)(z + b)
=

k∑
`=0

〈
−

1

2

〉
`

1

(ab)`+1/2
2`Bk,`

(
a + b

2
, 1, 0, . . . , 0

)

=

k∑
`=0

〈
−

1

2

〉
`

1

(ab)`+1/2
2`

(k − `)!
2k−`

(
k

`

)(
`

k − `

)(
a + b

2

)2`−k
.

Letting z → 0 on both sides of (2.3), employing the above result, and simplifying lead to∫ b
a

1√
(t−a)(b− t)

1

tk+1
d t

=
(−1)kπ

(a + b)k
√
ab

k∑
`=0

(−1)`22`
(2`− 1)!!

(2`)!!

(
`

k − `

)(
a + b

2

)`(
1/a + 1/b

2

)`
,

that is,

I(a,b; 0,k + 1) =
π

G(a,b)

(−1)k

[2A(a,b)]k

k∑
`=0

(−1)`22`
(2`− 1)!!

(2`)!!

(
`

k − `

)[
A(a,b)

H(a,b)

]`
(2.6)

for b > a > 0 and k ≥ 0, where
(
p
q

)
= 0 for q > p ≥ 0, the double factorial of negative odd integers

−(2n + 1) is defined by

(−2n− 1)!! =
(−1)n

(2n− 1)!!
= (−1)n

2nn!

(2n)!
, n = 0, 1, . . . ,

and the quantities

A(a,b) =
a + b

2
, G(a,b) =

√
ab, and H(a,b) =

2
1
a

+ 1
b

are respectively the well-known arithmetic, geometric, and harmonic means of two positive numbers a
and b.

When k = 0 in (2.6), the integral (1.2) or (2.1) is recovered.
In fact, the above argument implies that∫ b
a

1√
(t−a)(b− t)

1

(t + z)k+1
d t =

(−1)k

[2A(z + a,z + b)]k
π

G(z + a,z + b)

×
k∑
`=0

(−1)`22`
(2`− 1)!!

(2`)!

(
`

k − `

)[
A(z + a,z + b)

H(z + a,z + b)

]`
for b > a > 0 and k ≥ 0. This is equivalent to (2.6).

By the way, the ratio
(2`−1)!!

(2`)!
is called the Wallis ratio. For more information, please refer to the

paper [7] and plenty of references cited therein.
Alternatively differentiating with respect to z on both sides of (1.1) leads to

dk

d zk
1√

(z + a)(z + b)
=

dk

d zk

(
1

√
z + a

1
√
z + b

)

=

k∑
`=0

(
k

`

)(
1

√
z + a

)(`)(
1

√
z + b

)(k−`)



106 QI AND ČERŇANOVÁ

=

k∑
`=0

(
k

`

)〈
−

1

2

〉
`

1

(z + a)`+1/2

〈
−

1

2

〉
k−`

1

(z + b)k−`+1/2

=

k∑
`=0

(
k

`

)
(−1)`

(2`− 1)!!
2`

1

(z + a)`+1/2
(−1)k−`

[2(k − `) − 1]!!
2k−`

1

(z + b)k−`+1/2

=
(−1)k

2k
1

(z + a)1/2
1

(z + b)k+1/2

k∑
`=0

(
k

`

)
(2`− 1)!![2(k − `) − 1]!!

(
z + b

z + a

)`
.

Substituting this into (2.3) and taking the limit z → 0 result in

I(a,b; 0,k + 1) =
π
√
ab

1

bk

k∑
`=0

(2`− 1)!!
(2`)!!

[2(k − `) − 1]!!
[2(k − `)]!!

(
b

a

)`
for b > a > 0 and k ≥ 0. This is an alternative expression for I(a,b; 0,k + 1).

2.7. Under different conditions from those discussed above on b > a > 0 and λ,η ∈ R, can one discover
more explicit formulas for the improper integral I(a,b; λ,η)?

3. Identities for I(a,b; k,η)

In this section, we present several identities for the improper integral I(a,b; k,η).

3.1. Substituting s = 1
t

into I(a,b; k,η) yields

I(a,b; k,η) =
(−1)k
√
ab

I

(
1

b
,

1

a
; k, 1 −η

)
(3.1)

for k ≥ 0, η ∈ R, and a,b > 0 with a 6= b. In particular, it can be derived that

I(a,b; 0, 1) =
1
√
ab
I

(
1

b
,

1

a
; 0, 0

)
and

I

(
1

b
,b; k,η

)
= (−1)kI

(
1

b
,b; k, 1 −η

)
.

3.2. Substituting s = t
a

into I(a,b; k,η) gives

I(a,b; k,η) =
1

aη

[(
lnk a

)
I

(
1,
b

a
; 0,η

)
+ I

(
1,
b

a
; k,η

)]
for k ∈ N, η ∈ R, and a,b > 0 with a 6= b. In particular,

I(a, 1; k,η) =
1

aη

[(
lnk a

)
I

(
1,

1

a
; 0,η

)
+ I

(
1,

1

a
; k,η

)]
. (3.2)

3.3. From (3.1), it follows that

I(a, 1; k,η) =
(−1)k
√
a

I

(
1,

1

a
; k, 1 −η

)
(3.3)

Substituting (3.3) into (3.2) leads to

I

(
1,

1

a
; k,η

)
=

(−1)k

aη−1/2
I

(
1,

1

a
; k, 1 −η

)
−
(
lnk a

)
I

(
1,

1

a
; 0,η

)
for 1 6= a > 0, k ∈ N, and η ∈ R. Consequently,

I(1,b; k,η) =
(−1)k

b1/2−η
I(1,b; k, 1 −η) +

(
lnk b

)
I(1,b; 0,η)



SOME DISCUSSIONS ON A KIND OF IMPROPER INTEGRALS 107

for 1 6= b > 0, k ∈ N, and η ∈ R.

4. Remarks

By the way, we list two remarks on (1.1) and integral representations of the weighted geometric
means.

Remark 4.1. The integral representation (1.1) can be generalized as follows. For ak < ak+1 and
wk > 0 with

∑n
k=1 wk = 1, the principal branch of the reciprocal of the weighted geometric mean∏n

k=1(z + ak)
wk on C\ (−∞,−a1] can be represented by

1∏n
k=1(z + ak)

wk
=

1

π

n−1∑
m=1

sin

(
π

m∑
`=1

w`

)∫ am+1
am

1∏n
k=1 |t−ak|wk

1

t + z
d t.

Remark 4.2. Before getting the integral representation (1.1), the following integral representation
for the weight geometric mean

∏n
k=1(z + ak)

wk was obtained. Let wk > 0 and
∑n
k=1 wk = 1 for

1 ≤ k ≤ n and n ≥ 2. If a = (a1,a2, . . . ,an) is a positive and strictly increasing sequence, that is,
0 < a1 < a2 < · · · < an, then the principal branch of the weighted geometric mean

Gw,n(a + z) =

n∏
k=1

(ak + z)
wk, z ∈ C\ (−∞,−a1]

has the Lévy–Khintchine expression

Gw,n(a + z) = Gw,n(a) + z +

∫ ∞
0

ma,w,n(u)(1 −e−zu) d u, (4.1)

where the density

ma,w,n(u) =
1

π

n−1∑
`=1

sin

(
π
∑̀
j=1

wj

)∫ a`+1
a`

n∏
k=1

|ak − t|wke−ut d t.

For more detailed information, please refer to [1, 6, 8, 9, 12, 13, 14, 15, 16] and closely-related references
therein.

Remark 4.3. Letting n = 2 and w1 = w2 =
1
2

in (4.1) or setting n = 2 in [14, Theorem 1.1] leads to

√
(z + a)(z + b) =

√
ab + z +

1

π

∫ ∞
0

[∫ b
a

√
(b− t)(t−a) e−ut d t

]
(1 −e−zu) d u

=
√
ab + z +

1

π

∫ b
a

√
(b− t)(t−a)

[∫ ∞
0

e−ut(1 −e−zu) d u
]

d t

=
√
ab + z +

z

π

∫ b
a

√
(b− t)(t−a)

t

1

t + z
d t,

that is, ∫ b
a

√
(b− t)(t−a)

t

1

t + z
d t = π

[√
(z + a)(z + b) −

√
ab

z
− 1
]
,

for b > a > 0. Taking the limit z → 0 on both sides of (4.2) yields∫ b
a

√
(b− t)(t−a)

t2
d t = π

(
a + b

2
√
ab
− 1
)

= π

[
A(a,b)

G(a,b)
− 1
]
, b > a > 0. (4.2)

For k ∈ N, differentiating k times with respect to z procures

1

π

∫ b
a

√
(b− t)(t−a)

t

(−1)kk!
(t + z)k+1

d t =

[√
(z + a)(z + b) −

√
ab

z

](k)
=
√
ab

[
1

z

(√
1 +

a + b

ab
z +

1

ab
z2 − 1

)](k)



108 QI AND ČERŇANOVÁ

=
√
ab

[
1

z

∞∑
`=1

〈
1

2

〉
`

1

`!

(
a + b

ab
z +

1

ab
z2
)`](k)

,

∣∣∣∣a + bab z + 1abz2
∣∣∣∣ < 1

=
√
ab

∞∑
`=1

(−1)`−1
(2`− 3)!!

2`
1

`!

1

(ab)`
[
z`−1(a + b + z)`

](k)
=
√
ab

∞∑
`=1

(−1)`−1
(2`− 3)!!

2`
1

`!

1

(ab)`

k∑
q=0

(
k

q

)(
z`−1

)(q)[
(a + b + z)`

](k−q)
→
√
ab

k+1∑
`=1

(−1)`−1
(2`− 3)!!

2`
1

`!

1

(ab)`

(
k

`− 1

)
(`− 1)! lim

z→0

[
(a + b + z)`

](k−`+1)
=
√
ab

k+1∑
`=1

(−1)`−1
(2`− 3)!!

2`
1

`

1

(ab)`

(
k

`− 1

)
〈`〉k−`+1(a + b)2`−k−1

=
1

(a + b)k−1
√
ab

k∑
`=0

(−1)`
(2`− 1)!!

2`+1
1

` + 1

(
k

`

)
〈` + 1〉k−`

(a + b)2`

(ab)`

=
1

(a + b)k−1
√
ab

k∑
`=0

(−1)`
(2`− 1)!!

2`+1
1

` + 1

(
k

`

)
(` + 1)!

(2`−k + 1)!
(a + b)2`

(ab)`

=
k!

(a + b)k−1
√
ab

k∑
`=0

(−1)`
(2`− 1)!!

[2(` + 1)]!!

(
` + 1

k − `

)
(a + b)2`

(ab)`

as z → 0. As a result, we have∫ b
a

√
(b− t)(t−a)

tk+2
d t = π

(−1)k

(a + b)k

k∑
`=0

(−1)`
(2`− 1)!!
[2(` + 1)]!!

(
` + 1

k − `

)(
a + b
√
ab

)2`+1
for b > a > 0 and k ∈ N.

Remark 4.4. This paper is a slightly modified version of the preprint [10].

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1Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia
Autonomous Region, 028043, China

3Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387,
China

4Institute of Computer Science and Mathematics, Slovak University of Technology, Bratislava, Slovak
Republic

∗Corresponding author: qifeng618@gmail.com