EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 992-998
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Triple Integral Containing the Lommel Function
su,v(z): Derivation and Evaluation

Robert Reynolds1,∗, Allan Stauffer1

1 Department of Mathematics and Statistics, Faculty of Science, York University, Toronto,
Ontario, Canada, M3J1P3

Abstract. A three-dimensional integral containing the kernel g(x, y, z)su,v(z) is derived. The
function g(x, y, z) is a generalized function containing the logarithmic and exponential functions
and su,v(z) is the Lommel function and the integral is taken over the cube 0 ≤ y ≤ ∞, 0 ≤ x ≤
∞, 0 ≤ z ≤ ∞. A representation in terms of the Lerch function is derived, from which special cases
can be evaluated. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distribution.
All the results in this work are new.

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

Key Words and Phrases: Lommel Function, Triple integral, Catalan’s constant, Cauchy integral

1. Significance Statement

Eugen Cornelius Joseph von Lommel (1837-1899) was a German physicist. He is known
for the Lommel polynomial, the Lommel function, the Lommel-Weber function, and the
Lommel differential equation. The Lommel function given in equation (10.7.10) in [10] is
a particular solution to the inhomogeneous Bessel equation given in equation (10.7.4) in
[10]. These functions see a myriad of uses in physics and engineering (see [3] for a complete
list of references). Definite integrals in the form of Mellin transforms of Lommel functions
are tabled in the book of [1]. In this work the authors extend the dimension of the integral
and the kernel involving the Lommel function. In this paper the authors derive a triple
integral of the product of the Lommel, logarithmic and exponential functions and express
this triple integral in terms of the Hurwitz-Lerch Zeta function Φ(z, s, v).

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4282

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

https://www.ejpam.com 992 © 2022 EJPAM All rights reserved.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 992-998 993

2. Introduction

In this paper we derive the triple definite integral given by

∫ ∞

0

∫ ∞

0

∫ ∞

0

αxme−b(y2+z2)y−m−v+1z−m+v+1(αx)u logk
(
ax
yz

)
u2 + 2u− v2 + 1

1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−1

4
x2α2

)
dxdydz (1)

where the parameters k, a, α are general complex numbers and −1 < Re(m) < Re(v) <
Re(u) < 1, Re(b) > 0, Re(α) > 0. 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 [8]. 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 z, 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 and 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 [8, 9]. The variable of integration in the contour integral is
r = w + m. The cut and contour are in the first or second quadrant of the complex
r-plane. 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

(
ax

yz

)
and multiplying by

αxme−b(y2+z2)y−m−v+1z−m+v+1(αx)u 1F2

(
1; u2 − v

2 + 3
2 ,

u
2 + v

2 + 3
2 ;−

1
4x

2α2
)

u2 + 2u− v2 + 1

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

1

Γ(k + 1)

∫ ∞

0

∫ ∞

0

∫ ∞

0

αxme−b(y2+z2)y−m−v+1z−m+v+1(αx)u logk
(
ax
yz

)
u2 + 2u− v2 + 1



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 992-998 994

1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−1

4
x2α2

)
dxdydz

=
1

2πi

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫
C

αaww−k−1e−b(y2+z2)xm+w(αx)uy−m−v−w+1z−m+v−w+1

u2 + 2u− v2 + 1

1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−1

4
x2α2

)
dwdxdydz

=
1

2πi

∫
C

∫ ∞

0

∫ ∞

0

∫ ∞

0

αaww−k−1e−b(y2+z2)xm+w(αx)uy−m−v−w+1z−m+v−w+1

u2 + 2u− v2 + 1

1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−1

4
x2α2

)
dxdydzdw

=
1

2πi

∫
C
πaww−k−1bm+w−2

(
−2m+u+w−4

)
α−m−w

Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
csc

(
1

2
π(m+ u+ w − 1)

)
dw (3)

from equation (3.37.5.1) in [1] and equations (3.326.3) and (8.574.3) in [4] where Re(α) >
0, |Re

(
1
2π(m+ u+ w − 1)

)
|< 1, Re(w+m) < 3/2 and using the reflection formula (8.334.3)

in [4] for the Gamma function. We are able to switch the order of integration over x, y, z
and r using Fubini’s theorem for multiple integrals see (9.112) in [5], 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 [2] has a series representation given by

Φ(z, s, v) =
∞∑
n=0

(v + n)−szn (4)

where |z|< 1, v ̸= 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 ̸= 1, Re(s) > 0, or z = 1, Re(s) > 1.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 992-998 995

4.2. Infinite sum of the Contour Integral

Using equation (2) and replacing y by

log(a)− log(α) + log(b) +
1

2
iπ(2y + 1) + log(2)

then multiplying both sides by

π(−1)ybm−2α−m2m+u−3e
1
2
iπ(2y+1)(m+u)Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

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

1

Γ(k + 1)
πk+1bm−2α−m2m+u−3e

1
2
iπ(k+m+u)Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
Φ

(
−eiπ(m+u),−k,

−2i log(2a)− 2i log(b) + 2i log(α) + π

2π

)

=
1

2πi

∞∑
y=0

∫
C
πaww−k−1bm+w−22m+u+w−3α−m−w

Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
e

1
2
iπ(2y(m+u+w+1)+m+u+w)dw

=
1

2πi

∫
C

∞∑
y=0

πaww−k−1bm+w−22m+u+w−3α−m−w

Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
e

1
2
iπ(2y(m+u+w+1)+m+u+w)dw

=
1

2πi

∫
C
πaww−k−1bm+w−22m+u+w−4α−m−w

Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
sec

(
1

2
π(m+ u+ w)

)
dw (6)

from equation (1.232.2) in [4] where Im
(
1
2π(m+ u+ 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,−1 < Re(m) < Re(v) < Re(u) < 1, Re(b) > 0, Re(α) > 0
then,



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 992-998 996

∫ ∞

0

∫ ∞

0

∫ ∞

0

αxme−b(y2+z2)y−m−v+1z−m+v+1(αx)u logk
(
ax
yz

)
u2 + 2u− v2 + 1

1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−1

4
x2α2

)
dxdydz

= πk+1bm−2α−m2m+u−3e
1
2
iπ(k+m+u)Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
Φ

(
−eiπ(m+u),−k,

−2i log(2a)− 2i log(b) + 2i log(α) + π

2π

)
(7)

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.

∫ ∞

0

∫ ∞

0

∫ ∞

0

αxme−b(y2+z2)y−m−v+1z−m+v+1(αx)u

u2 + 2u− v2 + 1

1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−1

4
x2α2

)
dxdydz

= πbm−2α−m2m+u−4 sec

(
1

2
π(m+ u)

)
Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
(8)

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

Example 2. The inverse tangent function tan−1(x),∫ ∞

0

∫ ∞

0

∫ ∞

0

e−y2−z2xm+uy−m−v+1z−m+v+1

(u2 + 2u− v2 + 1) log
(
− x

2yz

)
1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−x2

4

)
dxdydz

= 2m+u−2e−
1
2
iπ(2m+2u+1)

(
e

1
2
iπ(m+u) − tan−1

(
e

1
2
iπ(m+u)

))
Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
(9)

Proof. Use equation (7) and set k = −1, a = −1/2, b = 1, α = 1 and simplify using
entry (3) in Table below (64:12:7) in [7].



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (3) (2022), 992-998 997

Example 3. The inverse hyperbolic tangent function tanh−1(x),

∫ ∞

0

∫ ∞

0

∫ ∞

0

x10/21y17/28z31/28e−y2−z2
1F2

(
1; 3724 ,

43
24 ;−

x2

4

)
log

(
− x

2yz

) dxdydz

= − 42
√
−1210/21

(
(−1)5/21 + i tanh−1

(
(−1)31/42

))
Γ

(
37

24

)
Γ

(
43

24

)
(10)

Proof. Use equation (9) and set u = 1/3, v = 1/4,m = 1/7 and simplify.

Example 4. The Polylogarithm function Lik(z),

∫ ∞

0

∫ ∞

0

∫ ∞

0

e−y2−z2xm+uy−m−v+1z−m+v+1 logk
(

ix
2yz

)
u2 + 2u− v2 + 1

1F2

(
1;

u

2
− v

2
+

3

2
,
u

2
+

v

2
+

3

2
;−x2

4

)
dxdydz

= πk+1
(
−2m+u−3

)
e

1
2
iπ(k+m+u)−iπ(m+u)

Γ

(
1

2
(u− v + 1)

)
Γ

(
1

2
(u+ v + 1)

)
Li−k

(
−eiπ(m+u)

)
(11)

Proof. Use equation (7) and set a = i/2, b = 1, α = 1 and simplify using equation
(64:12:2) in [7].

Example 5. The constant π,

∫ ∞

0

∫ ∞

0

∫ ∞

0

y7/6z11/6e−y2−z2
1F2

(
1; 1912 ,

23
12 ;−

x2

4

)
log2

(
ix
2yz

) dxdydz = −
77πΓ

(
7
12

)
Γ
(
11
12

)
3456

(12)

Proof. Use equation (11) and set k = −2, u = 1/2, v = 1/3,m = −1/2 and simplify.

Example 6. Catalan’s constant K,

∫ ∞

0

∫ ∞

0

∫ ∞

0

√
xy2/3z4/3e−y2−z2

1F2

(
1; 1912 ,

23
12 ;−

x2

4

)
log2

(
ix
2yz

) dxdydz

= −
(

77
13824 − 77i

13824

) (
π2 + 48iK

)
Γ
(

7
12

)
Γ
(
11
12

)
π

(13)

Proof. Use equation (11) and set k = −2, u = 1/2, v = 1/3,m = 0 and simplify using
equation (2.2.1.2.7) in [6].



REFERENCES 998

6. Discussion

In this paper, we have presented a novel method for deriving a new triple integral
containing the Lommel function su,v(z) along with 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.

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