EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 437-442
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Triple Integral involving the Struve function HHHv(t)
expressed in terms of the Hurwitz-Lerch zeta Function

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

Abstract. The main focus of the present paper is to establish a triple integral involving the Struve
function in terms of the Hurwitz-Lerch zeta function by using our contour integral method. We
also consider some useful examples as special cases of this integral involving the Struve function
to give the applications of our main results.
2020 Mathematics Subject Classifications: 30E20, 33-01, 33-03, 33-04, 33-33B
Key Words and Phrases: Struve function, Triple integral, Hurwitz-Lerch zeta function

1. Significance Statement

The Struve function HHHv(t) was introduced by Hermann von Struve [7] and today this
function is carrying his name. This function is related to the non-homogeneous Bessel
type ordinary differential equation of special type called Struve differential equation [1].
The Struve function has a variety of applications, such as describing the vibrations of thin
disks and in the theory of electromagnetism, see section (57:1) in [3]. Other applications
of Struve functions are detailed in section (11.12) in [1]. In this work we use our contour
integral method to derive a triple integral involving the Struve function and expressed in
terms of the Hurwitz-Lerch zeta function. Multiple integrals involving special functions
are used to solve ordinary differential equations [9] and our goal is to provide an expansion
of such work.

2. Introduction

In this paper we derive the triple definite integral given by

(1)
∫ ∞

0

∫ ∞

0

∫ ∞

0
y1−mtm−v−1x−m+2v+1HHHv(t)e

−bx2−cy2 logk
(
at

xy

)
dxdydt

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4227

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

https://www.ejpam.com 437 © 2022 EJPAM All rights reserved.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 437-442 438

where the parameters k, a, b, c are general complex numbers and Re(m) < Re(v) + 3/2.
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 [4]. 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 and take the infinite sums 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 [4]. The variable of integration in the contour integral is β =
w +m. The cut and contour are in the second quadrant of the complex beta-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 form the triple integral by replacing y by log

(
at
xy

)
and

multiplying by y1−mtm−v−1x−m+2v+1HHHv(t)e
−bx2−cy2 then taking the definite integral with

respect to x ∈ [0,∞), y ∈ [0,∞) and t ∈ [0,∞) to obtain

(3)

1

Γ(k + 1)

∫ ∞

0

∫ ∞

0

∫ ∞

0
y1−mtm−v−1x−m+2v+1HHHv(t)e

−bx2−cy2 logk
(
at

xy

)
dxdydt

=
1

2πi

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫
C
aww−k−1y−m−w+1HHHv(t)

e−bx2−cy2tm−v+w−1x−m+2v−w+1dwdxdydt

=
1

2πi

∫
C

∫ ∞

0

∫ ∞

0

∫ ∞

0
aww−k−1y−m−w+1HHHv(t)

e−bx2−cy2tm−v+w−1x−m+2v−w+1dxdydtdw

=
1

2πi

∫
C
πaww−k−1c

m+w
2

−12m−v+w−3 sec

(
1

2
π(m+ w)

)
b
m+w

2
−v−1dw

from equation (13.2) on page 392 in [8] and equation (3.326.2) in [2] where −1 < Re(w +
m) < 0 and Re(w + m) < Re(v) + 3/2, Re(b) > 0, Re(c) > 0 and using the reflection
formula (8.334.3) in [2] 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,∞)



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 437-442 439

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 [1] and [5, 6] 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.

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 πc
m
2
−12m−v−2b

m
2
−v−1 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+1c

m
2
−1e

1
2
iπ(k+m)2m−v−2b

m
2
−v−1

Φ

(
−eimπ,−k,

−2i log(2a)− i log(b)− i log(c) + π

2π

)
=

1

2πi

∞∑
y=0

∫
C
π(−1)yaww−k−1c

1
2
(m+w−2)2m−v+w−2

e
1
2
iπ(2y+1)(m+w)b

1
2
(m−2v+w−2)dw

=
1

2πi

∫
C

∞∑
y=0

π(−1)yaww−k−1c
1
2
(m+w−2)2m−v+w−2

e
1
2
iπ(2y+1)(m+w)b

1
2
(m−2v+w−2)dw

=
1

2πi

∫
C
πaww−k−1c

1
2
(m+w−2)2m−v+w−3 sec

(
1

2
π(m+ w)

)
b
1
2
(m−2v+w−2)dw

from equation (1.232.2) in [2] where Im(w +m) > 0 in order for the sum to converge.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 437-442 440

5. Definite Integral in terms of the Lerch function

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

(7 )

∫ ∞

0

∫ ∞

0

∫ ∞

0
y1−mtm−v−1x−m+2v+1HHHv(t)e

−bx2−cy2 logk
(
at

xy

)
dxdydt

= πk+1c
m
2
−1e

1
2
iπ(k+m)2m−v−2b

m
2
−v−1

Φ

(
−eimπ,−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
y1−mtm−v−1x−m+2v+1HHHv(t)e

−bx2−cy2dxdydt

= πc
m
2
−12m−v−3 sec

(πm
2

)
b
m
2
−v−1

Proof. Use equation (7) and set k = 0 and simplify using entry (2) in Table below
(64:12:7) in [3].

Example 2. An example in terms of the Riemann zeta function ζ(k).∫ ∞

0

∫ ∞

0

∫ ∞

0
yt−v−1x2v+1e−x2−y2HHHv(t) log

k

(
it

2xy

)
dxdydt

=
(
2k+1 − 1

)
e

iπk
2 πk+1

(
−2−v−2

)
ζ(−k) (9)

Proof. Use equation (7) and set m = 0 and simplify in terms of the Hurwitz zeta
function ζ(k, a) using entry (4) in Table below (64:12:7) in [3]. Next set a = i/2, b = c = 1
and simplify in terms of the Riemann zeta function using entry (2) in Table below (64:7)
in [3].

Example 3.

(10)
∫ ∞

0

∫ ∞

0

∫ ∞

0

yt−v−1x2v+1e−x2−y2HHHv(t)

log
(

it
2xy

) dxdydt = −i2−v−2 log(2)



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 437-442 441

Proof. Use equation (9) and apply l’Hopital’s rule to the right-hand side as k → −1
and simplify using equation (25.6.11) in [1].

Example 4.

(11)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x2y(1− cos(t))e−x2−y2

t2
(
4 log2

(
t

2xy

)
+ π2

) dxdydt =
log(2)

16
√
π

and

(12)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x2y(cos(t) + 1)e−x2−y2 log
(

t
2xy

)
t2
(
4 log2

(
t

2xy

)
+ π2

) dxdydt = 0

Proof. Use equation (10) and set v = 1/2 and rationalize the denominator and equate
real and imaginary parts and simplify.

Example 5.

(13)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x7yHHH3(t)e
−x2−y2

t4
(
4 log2

(
t

2xy

)
+ π2

)dxdydt = log(2)

64π

and

(14)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x7yHHH3(t)e
−x2−y2 log

(
t

2xy

)
t4
(
log2

(
t

2xy

)
+ π2

4

) dxdydt = 0

Proof. Use equation (10) and set v = 3 and rationalize the denominator and equate
real and imaginary parts and simplify.

Example 6.

(15)
∫ ∞

0

∫ ∞

0

∫ ∞

0

xyHHH0(t)e
−x2−y2

t
(
4 log2

(
t

2xy

)
+ π2

)dxdydt = log(2)

8π

and

(16)
∫ ∞

0

∫ ∞

0

∫ ∞

0

xyHHH0(t)e
−x2−y2 log

(
t

2xy

)
t
(
4 log2

(
t

2xy

)
+ π2

) dxdydt = 0

Proof. Use equation (10) and set v = 0 and rationalize the denominator and equate
real and imaginary parts and simplify.

Example 7.

(17)
∫ ∞

0

∫ ∞

0

∫ ∞

0

x3yHHH1(t)e
−x2−y2

t2
(
4 log2

(
t

2xy

)
+ π2

)dxdydt = log(2)

16π

and

(18)
∫ ∞

0

∫ ∞

0

∫ ∞

0

4x3yHHH1(t)e
−x2−y2 log

(
t

2xy

)
t2
(
4 log2

(
t

2xy

)
+ π2

) dxdydt = 0



REFERENCES 442

Proof. Use equation (10) and set v = 1 and rationalize the denominator and equate
real and imaginary parts and simplify.

6. Discussion

In this paper, we have presented a novel method for deriving a new integral transform
in terms of the Struve function along with some interesting definite integrals with many
more possible, 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.

References

[1] Nist digital library of mathematical functions. 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.

[2] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Else-
vier/Academic Press, Amsterdam, seventh edition, 2007.

[3] Keith B. Oldham, Jan Myland, and Jerome Spanier. An Atlas of Functions: with
Equator, the Atlas Function Calculator. Springer Science & Business Media, 07 2010.

[4] 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.

[5] H. Srivastava. The zeta and related functions: Recent developments. J. Adv. Eng.
Comput., 2019.

[6] H.M. Srivastava. Some general families of the hurwitz-lerch zeta functions and their
applications: recent developments and directions for further researches.

[7] Hermann Struve. Beitrag zur theorie der diffraction an fernröhren. Annalen der Physik,
253:1008–1016, 1882.

[8] George N Watson. A treatise on the theory of Bessel functions. Cambridge Univ.
Press, 2011.

[9] Árpád Baricz, Dragana Jankov Maširevič, and Tibor K Pogány. Series of Bessel and
kummer-type functions. Springer, 2018.