EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 980-988
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Definite Integral of Power and Algebraic Functions in
terms of the Lerch Function

Robert Reynolds1,∗, Allan Stauffer1

1 Department of Mathematics and Statistics, Faculty of Science, York University, Toronto,
Ontario, Canada, M3J 1P3

Abstract. Bierens de haan (1867) evaluated a definite integral involving the cotangent function
and this result was also listed in Gradshteyn and Ryzhik (2007). The objective of this present note
is to use this integral along with Cauchy’s integral formula to derive a definite logarithmic integral
in terms of the Lerch function. We will use this integral formula to produce a table of known and
new results in terms of special functions and thereby expanding the list of definite integrals in both
text books.

2020 Mathematics Subject Classifications: 30E20,33-01, 33-03, 33-04, 33-33B, 33E20,33E33

Key Words and Phrases: Entries in Bierens de Haan, divergent integral, Cauchy integral,
Catalan’s constant, Glaisher’s constant

1. Introduction

A thorough review of the Bierens de haan (1867) and Gradshteyn and Rhyzik’s (2007)
books of integral tables showcases a vast number of difficult and unknown integral formulas.
We shall derive and evaluate the integral

(1)

∫ ∞
0


(

bx
bx+1

)m
logk

(
abx
bx+1

)
x

−
b
(
bx+1
bx

)m
logk

(
a(bx+1)
bx

)
bx+ 1

 dx

where a, k, b and m are general complex numbers. This integral was of particular interest
because it showcases two integrands which by themselves are divergent. However, if the two
integrands are the same at large values, the difference of the integrands gives a convergent
integral. In this work we provide a formal derivation for known and new integrals and
tabulate these definite integrals in terms of special functions and fundamental constants.
This could be viewed as a new entry table for books with table of integral formulae such

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4017

Email addresses: milver@my.yorku.ca (R. Reynolds), stauffer@yorku.ca (A. Stauffer)

http://www.ejpam.com 980 c© 2021 EJPAM All rights reserved.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 14 (3) (2021), 980-988 981

as [7], [8], [12] and [4]. The derivations follow the method used by us in [10], [11] and [9].
This method involves using a form of the generalized Cauchy’s integral formula given by

yk

k!
=

1

2πi

∫
C

ewy

wk+1
dy. (2)

where C is in general an open contour in the complex plane where the bilinear concomi-
tant [11] has the same value at the end points of the contour. Then we multiply both sides
by a function, then take a definite integral of both sides. This yields a definite integral
in terms of a contour integral. Then we multiply both sides of equation (2) by another
function and take the infinite sum of both sides such that the contour integral of both
equations are the same.

Definite Integral of the Contour Integral

We use the method in [11]. The variable of integration in the contour integral is
z = m+w. The cut and contour are in the second quadrant of the complex z-plane. The
cut approaches the origin from the interior of the second quadrant and the contour goes
round the origin with zero radius and is on opposite sides of the cut. Using a generalization

of Cauchy’s integral formula we first replace y by log
(
abx
bx+1

)
then multiply by

( bx
bx+1)

m

x for

the first equation and then y by log
(
a(bx+1)
bx

)
and multiply by

b( bx+1
bx )

m

bx+1 to get the second

equation. Then we subtract these two equations to get

(3)

1

k!

∫ ∞
0


(

bx
bx+1

)m
logk

(
abx
bx+1

)
x

−
b
(
bx+1
bx

)m
logk

(
a(bx+1)
bx

)
bx+ 1

 dx

=
1

2πi

∫ ∞
0

∫
C

aww−k−1
(

bx
bx+1

)m+w

x
−
baww−k−1

(
bx+1
bx

)m+w

bx+ 1

 dwdx

=
1

2πi

∫
C

∫ ∞
0

aww−k−1
(

bx
bx+1

)m+w

x
−
baww−k−1

(
bx+1
bx

)m+w

bx+ 1

 dxdw

=
1

2πi

∫
C
πaww−k−1 cot(π(m+ w))dw

from Eq (3.217) in [12], where 0 < Re(m+ w) < 1 and Re(b) > 0, where the logarithmic
function is defined in equation (4.1.2) in [1].

Definition of the Lerch Function

The Lerch function has a series representation given by



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 14 (3) (2021), 980-988 982

Φ(z, s, v) =
∞∑
n=0

(v + n)−szn (4)

where |z|< 1, v 6= 0,−1, .. and is continued analytically by its integral representation given
by

Φ(z, s, v) =
1

Γ(s)

∫ ∞
0

ts−1e−vt

1− ze−t
dt =

1

Γ(s)

∫ ∞
0

ts−1e−(v−1)t

et − z
dt (5)

where Re(v) > 0, and either |z|≤ 1, z 6= 1, Re(s) > 0, or z = 1, Re(s) > 1.

Infinite Sum of the Contour Integral

In this section we will again use Cauchy’s integral formula (2) and taking the infinite
sum to derive equivalent sum representations for the contour integrals. First we replace y
by log(a) + 2iπ(y + 1)) and multiply both sides by −2iπe2iπm(y+1) to get

(6)−
iik(2π)k+1e2iπmy+2iπm

(
− i log(a)

2π + y + 1
)k

k!

= − 1

2πi

∫
C

2iπw−k−1 exp(w(log(a) + 2iπ(y + 1)) + 2iπm(y + 1))dw

Next we take the infinite sum over y ∈ [0,∞) and simplify in terms of the Lerch function
to get

(7)

−
(2iπ)k+1e2iπmΦ

(
e2imπ,−k, 1− i log(a)

2π

)
k!

= −
∞∑
y=0

1

2πi

∫
C

(
2iπw−k−1 exp(w(log(a) + 2iπ(y + 1)) + 2iπm(y + 1))

)
dw

= − 1

2πi

∫
C

∞∑
y=0

(
2iπw−k−1 exp(w(log(a) + 2iπ(y + 1)) + 2iπm(y + 1))

)
dw

=
1

2πi

∫
C
πaww−k−1 cot(π(m+ w)) + iπaww−k−1dw

from Eq (1.232.1) in [12], where Im(m + w) > 0 for the sum to converge and we replace
w by −w + π/2.

The additional Contour Integral

Using Eq (2) and replacing y by using log(a) and multiply by πi to get

(8)
iπ logk(a)

k!
=

1

2πi

∫
C
iπaww−k−1dw



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 14 (3) (2021), 980-988 983

Definite Integral in terms of the Lerch Function

Since the right-hand side of Eq (3) is equal to the sum of Eq’s (7) and (8) we can
equate the left-hand sides to yield the definite integral given by

(9)

∫ ∞
0


(

bx
bx+1

)m
logk

(
abx
bx+1

)
x

−
b
(
bx+1
bx

)m
logk

(
a(bx+1)
bx

)
bx+ 1

 dx

= −(2iπ)k+1e2iπmΦ

(
e2imπ,−k, 1− i log(a)

2π

)
− iπ logk(a)

Table of Definite Integrals

In this section we use Eq (9) to derive a Table of definite integrals in terms of funda-
mental constants and special functions.

Derivation of entry 3.217 in [12]

Using Eq (9) replacing b by q, m by p and setting k = 0 simplifying we get

(10)

∫ ∞
0


(

qx
qx+1

)p
x

−
q
(

1
qx + 1

)p
qx+ 1

 dx = π cot(πp)

from entry (2) in Table below (64:12:7) in [6].

Derivation of new entry 3.217.1 in [12]

Using Eq (9) replacing b by q, m by p and setting k = a = 1 simplifying we get

(11)

∫ ∞
0

q
(

1
qx + 1

)p
qx+ 1

+

(
qx
qx+1

)p
x

 log

(
qx

qx+ 1

)
dx = −π2 csc2(πp)

from entry (1) in Table below (64:12:7) in [6].

Derivation of new entry 3.217.2 in [12]

Using Eq (9) replacing b by q and setting k = 2, a = 1 and m = 1/2 simplifying we get

(12)

∫ ∞
0


√

qx
qx+1 log2

(
qx
qx+1

)
x

−
q
√

qx+1
qx log2

(
qx+1
qx

)
qx+ 1

 dx = 0

from entry (2) in Table below (64:12:7) in [6].



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 14 (3) (2021), 980-988 984

Derivation of new entry 3.217.3 in [12]

Using Eq (9) setting k = 1, a = e, b = 1 and m = 1/2 simplifying we get

(13)

∫ ∞
0


√

1
x + 1

(
log
(

1
x + 1

)
+ 1
)

x+ 1
+
− log(x) + log(x+ 1)− 1

√
x
√
x+ 1

 dx = π2

from entry (3) in Table below (64:12:7) in [6].

Derivation of new entry 3.217.4 in [12]

Using Eq (9) and setting m = 1/2 and a = 1 simplifying we get

(14)

∫ ∞
0


√

bx
bx+1 logk

(
bx
bx+1

)
x

−
b
√

bx+1
bx logk

(
bx+1
bx

)
bx+ 1

 dx =
(

1− 2k+1
)

(2iπ)k+1ζ(−k)

from entries (2) in Table below (64:7) and entry (3) in Table below (64:12:7) in [6]

Derivation of new entry 3.217.5 in [12]

Using Eq (9) we first set a = −1 and m = 1/2 followed by taking the first partial
derivative with respect to k then setting k = 0 simplifying we get

∫ ∞
0


√

bx
bx+1 log

(
log
(

1
bx+1 − 1

))
x

−
log
(
log
(
− 1
bx − 1

))
x
√

1
bx + 1

 dx = 2iπ log

(
−

2Γ
(

1
4

)
Γ
(
−1

4

))
(15)

from equations (1.10.10) in [7], (25.14.2) in [2] and (64:13:3) in [6].

Derivation of new entry 3.217.6 in [12]

Using Eq (9) and setting a = 1 and simplifying we get

∫ ∞
0


(

bx
bx+1

)m
logk

(
bx
bx+1

)
x

−
b
(

1
bx + 1

)m
logk

(
1
bx + 1

)
bx+ 1

 dx = −(2iπ)k+1Li−k
(
e2imπ

)
(16)

from equation (1.11.14) in [7].



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 14 (3) (2021), 980-988 985

Derivation of new entry 3.217.7 in [12]

Using Eq (9) and setting k = −1 simplifying we get

(17)

∫ ∞
0


(

bx
bx+1

)m
x log

(
abx
bx+1

) − b
(

1
bx + 1

)m
(bx+ 1) log

(
a
bx + a

)
 dx

= −e2iπmΦ

(
e2imπ, 1, 1− i log(a)

2π

)
− iπ

log(a)

=
2π

2π − i log(a)
2F1

(
1, 1− i log(a)

2π
; 2− i log(a)

2π
; e2imπ

)
from Eq (9.559) in [12].

Derivation of new entry 3.217.8 in [12]

Using Eq (9) setting m = 1/2, k = −2 and a = 1− simplifying we get

(18)

∫ ∞
0


√

bx
bx+1

x log2
(

1
bx+1 − 1

) − 1

x
√

1
bx + 1 log2

(
− 1
bx − 1

)
 dx =

i(2G− 1)

π

where G is Catalan’s constant given by Eq (9.73) in [12].

Derivation of new entry 3.217.9 in [12]

Using Eq (9) we first set m = 1/2 and a = 1, then we take the first partial derivative
with respect to k to get

(19)

∫ ∞
0


√

bx
bx+1 log

(
log
(

bx
bx+1

))
logk

(
bx
bx+1

)
x

−
log
(
log
(

1
bx + 1

))
logk

(
1
bx + 1

)
x
√

1
bx + 1

 dx

= −2iπik
(

22k+1πk log(2)ζ(−k) +
(

2k+1 − 1
)

(2π)k
(
−ζ ′(−k) + log(2iπ)ζ(−k)

))
Next we apply L’Hopitals’ rule to the left-hand side as k → −1 simplifying to get

(20)

∫ ∞
0


√

bx
bx+1 log

(
log
(

bx
bx+1

))
x log

(
bx
bx+1

) −
log
(
log
(

1
bx + 1

))
x
√

1
bx + 1 log

(
1
bx + 1

)
 dx

=
1

2
log(2)

(
−2γ + iπ + log

(
8π2
))

where γ is Euler’s constant given by Eq (9.73) in [12].



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 14 (3) (2021), 980-988 986

Derivation of new entry 3.217.10 in [12]

Using Eq (19) and setting k = −2 simplifying we get

(21)

∫ ∞
0


√

bx
bx+1 log

(
log
(

bx
bx+1

))
x log2

(
bx
bx+1

) −
log
(
log
(

1
bx + 1

))
x
√

1
bx + 1 log2

(
1
bx + 1

)
 dx

=
1

48
iπ(−24 log(A) + 2γ − iπ + log(4))

where A is the Glaisher-Kinkelin constant given by Eq (2.15) in [3].



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 14 (3) (2021), 980-988 987

Summary of Results

In this section we generate a table of definite integrals which can be included in [12].

f(x)
∫∞

0 f(x)dx

(
qx
qx+1

)p
x −

q
(

1
qx

+1
)p

qx+1 π cot(πp)(
q
(

1
qx

+1
)p

qx+1 +

(
qx
qx+1

)p
x

)
log
(

qx
qx+1

)
−π2 csc2(πp)

√
qx
qx+1

log2
(

qx
qx+1

)
x −

q
√
qx+1
qx

log2
(
qx+1
qx

)
qx+1 0

√
1
x

+1(log( 1
x

+1)+1)
x+1 + − log(x)+log(x+1)−1√

x
√
x+1

π2

√
bx
bx+1

logk( bx
bx+1)

x −
b
√
bx+1
bx

logk( bx+1
bx )

bx+1

(
1− 2k+1

)
(2iπ)k+1ζ(−k)

√
bx
bx+1

log(log( 1
bx+1

−1))
x − log(log(− 1

bx
−1))

x
√

1
bx

+1
2iπ log

(
− 2Γ( 1

4)
Γ(− 1

4)

)
( bx
bx+1)

m
logk( bx

bx+1)
x − b( 1

bx
+1)

m
logk( 1

bx
+1)

bx+1 −(2iπ)k+1Li−k
(
e2imπ

)
( bx
bx+1)

m

x log( abx
bx+1)

− b( 1
bx

+1)
m

(bx+1) log( a
bx

+a)

2π 2F1

(
1,1− i log(a)

2π
;2− i log(a)

2π
;e2imπ

)
2π−i log(a)√

bx
bx+1

log(log( bx
bx+1))

x log( bx
bx+1)

− log(log( 1
bx

+1))

x
√

1
bx

+1 log( 1
bx

+1)
1
2 log(2)

(
−2γ + iπ + log

(
8π2
))

√
bx
bx+1

log(log( bx
bx+1))

x log2( bx
bx+1)

− log(log( 1
bx

+1))

x
√

1
bx

+1 log2( 1
bx

+1)
1
48 iπ(−24 log(A) + 2γ − iπ + log(4))

Discussion

In this paper we have derived a table of definite integrals known and new in terms of
special functions and fundamental constants. Table of definite integrals of the logarithmic
function provide a useful reference for research into various topics such as Perturbative
and Non-perturbative Aspects of Quantum Field Theory [5] etc. We have devoted this
note to provide an extended listing of such integrals in [4] and [12] for potential future
work. The present paper should be seen as an extension of these results.



REFERENCES 988

Conclusion

In this paper, we have presented a novel method for deriving some interesting definite
integrals using contour integration. The results presented were numerically verified for
both real and imaginary and complex values of the parameters in the integrals using
Mathematica by Wolfram.

References

[1] Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions with
formulas, graphs, and mathematical tables, 12 1972.

[2] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.2
of 2021-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider,
R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A.
McClain, eds.

[3] Steven R. Finch. Mathematical Constants. Cambridge University Press; Illustrated
edition (Aug. 18 2003), 08 2003.

[4] David Bierens de Haan. Nouvelles tables d’intgrales dfinies. P. Engels, 1867.

[5] H Latal and W Schweiger. Perturbative and Nonperturbative Aspects of Quantum
Field Theory. Springer Berlin Heidelberg, 1997.

[6] Keit Oldham, Jan Myland, and Jerome Spanier. An Atlas of Functions. Springer US,
2009.

[7] Bateman Manuscript Project, Harry Bateman, Arthur Erdélyi, United States, and
Office of Naval Research. Higher transcendental functions, volume 1. McGraw-Hill,
1953.

[8] A. B. Prudnikov. Integrals and Series. Routledge, 05 2018.

[9] Robert Reynolds and Allan Stauffer. A definite integral involving the logarithmic
function in terms of the lerch function. Mathematics, 7:1148, 11 2019.

[10] Robert Reynolds and Allan Stauffer. Definite integral of arctangent and polyloga-
rithmic functions expressed as a series. Mathematics, 7:1099, 11 2019.

[11] Robert Reynolds and Allan Stauffer. A method for evaluating definite integrals in
terms of special functions with examples. International Mathematical Forum, 15:235–
244, 2020.

[12] Daniel Zwillinger and Alan Jeffrey. Table of Integrals, Series, and Products. Academic
Press, 08 2000.