EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 390-396
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Triple Integral involving the Product of the
Logarithmic and Bessel Functions expressed in terms of

the Lerch Function
Robert Reynolds1,∗, Allan Stauffer1
1 Department of Mathematics and Statistics, Faculty of Science, York University, Toronto,
Ontario, Canada, M3J1P3

Abstract. The aim of the present document is to evaluate a triple integral involving the product
a general class of logarithmic, special and exponential functions. Importance of our results lies in
the fact that they involve the Bessel function of the First Kind, which is used in a wide range of
areas spanning Science and Engineering. Further we establish some special cases.
2020 Mathematics Subject Classifications: 30E20, 33-01, 33-03, 33-04, 33-33B
Key Words and Phrases: Triple integral, Bessel function, Catalan’s constant, Apéry’s constant,
Cauchy integral

1. Significance Statement

Triple integrals whose kernels feature special functions are tabled in the book of Prud-
nikov et al. [9], in evaluating Euler type integrals involving a general class of polynomials,
special functions and multivariable A-function [4], in the study of celestial mechanics or
Hamiltonian dynamics, as applied to the ellipsoidal components of galaxies [1], in the the-
ory of Eisentein series for the Group SL(3,R) and its applications to a binary problem,
and in the theory of automorphic forms, which are defined arithmetically on any reductive
Lie group, which have been studied intensively for many years [2].

Based on current literature triple integrals of Special functions is of high importance,
researched and used widely. One feature of current work on these integrals which is not
present is a closed form solution where possible. In our present work we derive a triple
integral whose kernel involves the Bessel function of the first kind Jv(t) and expressed it
in terms of the Hurwitz-Lerch zeta function. The Bessel function itself is a very important
function and are a set of solutions to a second-order differential equation that can appear
in a variety of contexts [6].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4203

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

https://www.ejpam.com 390 © 2022 EJPAM All rights reserved.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 390-396 391

2. Introduction

In this paper we derive the triple definite integral given by

(1)
∫ ∞

0

∫ ∞

0

∫ ∞

0
tmxv−my

1
2
(−m−v−1)Jv(t)e

−bx2−cy logk
(

at

x
√
y

)
dxdydt

where the parameters k, a, b, c, v,m are general complex numbers and Re(b) > 0, Re(c) >
0, Re(v) > 0, Re(m) < −1. This definite integral will be used to derive special cases in
terms of special functions and fundamental constants. The derivations follow the method
used by us in [10]. This method involves using a form of the generalized Cauchy’s integral
formula given by

yk

Γ(k + 1)
=

1

2πi

∫
C

ewy

wk+1
dw. (2)

where C is in general an open contour in the complex plane where the bilinear concomitant
has the same value at the end points of the contour. We then multiply both sides by a
function of x, y and t, then take a definite triple 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 of y take the infinite sum of both sides such that the contour integral of
both equations are the same.

3. Definite Integral of the Contour Integral

We use the method in [10]. The variable of integration in the contour integral is s =
w+m+v. The cut and contour are in the first or second quadrant of the complex s-plane
depending on the sign of s. The cut approaches the origin from the interior of the first or
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 form the triple
integral by replacing y by log

(
at

x
√
y

)
and multiplying by tmxv−my

1
2
(−m−v−1)Jv(t)e

−bx2−cy

then taking the definite triple integral with respect to x ∈ [0,∞) and y ∈ [0,∞) and



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 390-396 392

t ∈ [0,∞) to obtain

(3)

1

Γ(k + 1)

∫ ∞

0

∫ ∞

0

∫ ∞

0
tmxv−my

1
2
(−m−v−1)Jv(t)e

−bx2−cy

logk
(

at

x
√
y

)
dxdydt

=
1

2πi

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫
C
aww−k−1tm+wJv(t)e

−bx2−cyx−m+v−w

y
1
2
(−m−v−w−1)dwdxdydt

=
1

2πi

∫
C

∫ ∞

0

∫ ∞

0

∫ ∞

0
aww−k−1tm+wJv(t)e

−bx2−cyx−m+v−w

y
1
2
(−m−v−w−1)dxdydtdw

=
1

2πi

∫
C
πaww−k−12m+w−1b

1
2
(m−v+w−1)c

1
2
(m+v+w−1)

sec

(
1

2
π(m+ v + w)

)
dw

from equation (10.22.43) in [3] and (3.326.2) in [5] where Re(w+m+v) > −1, Re(w+m) <
−1/2 and using the reflection formula (8.334.3) in [5] for the Gamma function. We are
able to switch the order of integration over w, x, y and t using Fubini’s theorem since the
integrand is of bounded measure over the space C× [0,∞)× [0,∞)× [0,∞)

4. The Hurwitz-Lerch zeta Function and Infinite Sum of the Contour
Integral

In this section we use Equation (2) to derive the contour integral representations for
the Hurwitz-Lerch zeta function.

4.1. The Hurwitz-Lerch zeta Function

The Hurwitz-Lerch zeta function (25.14) in [3] has a series representation given by

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.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 390-396 393

4.2. Infinite sum of the Contour Integral

Using equation (2) and replacing y by log(a)+ log(b)
2 + log(c)

2 + 1
2 iπ(2y+1)+log(2) then

multiplying both sides by π2m(−1)yb
1
2
(m−v−1)c

1
2
(m+v−1)e

1
2
iπ(2y+1)(m+v) taking the infinite

sum over y ∈ [0,∞) and simplifying in terms of the Hurwitz-Lerch zeta function we obtain

(6)

1

Γ(k + 1)
πk+12mb

1
2
(m−v−1)c

1
2
(m+v−1)e

1
2
iπ(k+m+v)

Φ

(
−eiπ(m+v),−k,

−2i log(2a)− i log(b)− i log(c) + π

2π

)
=

1

2πi

∞∑
y=0

∫
C
π(−1)yaww−k−12m+wb

1
2
(m−v+w−1)c

1
2
(m+v+w−1)e

1
2
iπ(2y+1)(m+v+w)dw

=
1

2πi

∫
C

∞∑
y=0

π(−1)yaww−k−12m+wb
1
2
(m−v+w−1)c

1
2
(m+v+w−1)e

1
2
iπ(2y+1)(m+v+w)dw

=
1

2πi

∫
C
πaww−k−12m+w−1b

1
2
(m−v+w−1)c

1
2
(m+v+w−1) sec

(
1

2
π(m+ v + w)

)
dw

from equation (1.232.2) in [5] where Im
(
1
2π(m+ v + w)

)
> 0 in order for the sum to

converge.

5. Definite Integral in terms of the Hurwitz-Lerch zeta Function

Theorem 1. For all k, a ∈ C, Re(b) > 0, Re(c) > 0, Re(v) > 0, Re(m) < −1,

(7 )

∫ ∞

0

∫ ∞

0

∫ ∞

0
tmxv−my

1
2
(−m−v−1)Jv(t)e

−bx2−cy logk
(

at

x
√
y

)
dxdydt

= πk+12mb
1
2
(m−v−1)c

1
2
(m+v−1)e

1
2
iπ(k+m+v)

Φ

(
−eiπ(m+v),−k,

−2i log(2a)− i log(b)− i log(c) + π

2π

)
Proof. The right-hand sides of relations (3) and (6) are identical; hence, the left-hand

sides of the same are identical too. Simplifying with the Gamma function yields the desired
conclusion.

Example 1. The degenerate case.

(8)

∫ ∞

0

∫ ∞

0

∫ ∞

0
tmxv−my

1
2
(−m−v−1)Jv(t)e

−bx2−cydxdydt

= π2m−1b
1
2
(m−v−1)c

1
2
(m+v−1) sec

(
1

2
π(m+ v)

)



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 390-396 394

Proof. Use equation (7) and set k = 0 and simplify using entry (2) in Table below
(64:12:7) in [8].

Example 2.

(9)

∫ ∞

0

∫ ∞

0

∫ ∞

0

tme−x2−yxv−my
1
2
(−m−v−1)Jv(t)

log
(
− t

2x
√
y

) dxdydt

= 2m+1e−
1
2
iπ(2m+2v+1)

(
e

1
2
iπ(m+v) − tan−1

(
e

1
2
iπ(m+v)

))
Proof. Use equation (7) and set k = −1, a = −1/2, b = c = 1 and simplify using entry

(3) in Table below (64:12:7) in [8].

Example 3.

(10)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x3J 5
3
(t)e−x2−y

t4/3y2/3
(
log2

(
t

2x
√
y

)
+ π2

)dxdydt = −π +
√
3(log(3)− 4)

8
3
√
2π

and

(11)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x3J 5
3
(t)e−x2−y log

(
t

2x
√
y

)
t4/3y2/3

(
log2

(
t

2x
√
y

)
+ π2

)dxdtdt = −4 +
√
3π − log(3)

8
3
√
2

Proof. Use equation (9) and set m = −4/3, v = 5/3 rationalize the denominator and
simplify.

Example 4. The Polylogarithm function Lik(z),

(12)

∫ ∞

0

∫ ∞

0

∫ ∞

0
tme−x2−yxv−my

1
2
(−m−v−1)Jv(t) log

k

(
it

2x
√
y

)
dxdydt

= πk+1 (−2m)

e
1
2
iπ(k+m+v)−iπ(m+v)Li−k

(
−eiπ(m+v)

)
Proof. Use equation (7) and set a = i/2, b = c = 1 and simplify using equation

(64:12:2) in [8].

Example 5. Catalan’s constant G

(13)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x2J 5
4
(t)e−x2−y

t3/4y3/4 log2
(

it
2x

√
y

)dxdydt = (
−1

2

)3/4 (
π2 + 48iG

)
48π



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 390-396 395

Proof. Use equation (12) and set k = −2,m = −3/4, v = 5/4 and simplify using
equation (2.2.1.2.7) in [7].

Example 6.

(14)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x3/2J 3
4
(t)e−x2−y logk

(
it

2x
√
y

)
t3/4

√
y

dxdydt = −
(
2k+1 − 1

)
e

iπk
2 πk+1ζ(−k)

23/4

Proof. Use equation (12) and set m = −3/4, v = 3/4 and simplify using entry (2) in
Table below (64:7) in [8].

Example 7. The fundamental constant log(2),

(15)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x3/2J 3
4
(t)e−x2−y

t3/4
√
y log

(
it

2x
√
y

)dxdydt = − i log(2)

23/4

Proof. Use equation (14) apply l’Hopital’s rule as k → −1 and simplify.

Example 8. Apéry’s constant ζ(3)

(16)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x3/2J 3
4
(t)e−x2−y

t3/4
√
y log3

(
it

2x
√
y

)dxdydt = 3iζ(3)

4 23/4π2

Proof. Use equation (14) set k = −3 and simplify.

Example 9. The fundamental constant ζ(5),

(17 )
∫ ∞

0

∫ ∞

0

∫ ∞

0

x3/2J 3
4
(t)e−x2−y

t3/4
√
y log5

(
it

2x
√
y

)dxdydt = − 15iζ(5)

16 23/4π4

Proof. Use equation (14) set k = −5 and simplify.

6. Discussion

In this paper, we have presented a novel method for deriving a new Bessel function
integral transform along with some interesting definite integrals similar to those published
by Prudnikov et al. [9], 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.

Acknowledgements

This research is supported by NSERC Canada under grant 504070.



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