Interdisciplinary Application of Residue Theorem in Trigonometric
and Fraction Functions
Yeheng Xiao
a
School Malvern College Qingdao, Qingdao, China
Keywords: Residue Theorem, Complex Variable Function, Definite Integral.
Abstract: In complex function theory, there exists a fundamental concept known as the residue theorem., lies at the
intersection of pure mathematics and diverse scientific applications. This theorem elegantly connects the
integrand function over a precisely defined closed geometric contour in the complex plane to the sum of its
residues at isolated singularities within that contour. In mathematics, it serves as a potent means for the
evaluation of otherwise intractable complex integrals, offering new insights into the behavior of complex -
valued functions, from analyzing the distribution of zeros and poles to studying function singularities. Its
applications span multiple scientific disciplines. In physics, it simplifies calculations in quantum mechanics,
electromagnetism, and statistical physics, providing crucial solutions for problems like scattering amplitudes
and electromagnetic field distributions. In engineering, it aids in signal processing and control system design,
especially when dealing with Laplace and Fourier transforms. Hence, this paper aims to calculate
representative definite integrals with the help of Residue theorem, paving the way for connecting its
applications in interdisciplinary field.
1 INTRODUCTION
Complex variable functions are an important branch
in mathematics and are widely applied in many fields
such as physics, engineering, computer science, and
finance (Churchill & Brown, 2014). The following
are the main application scenarios of complex
variable functions. In the field of mathematics,
residue theorem relates closely to the complex
variable functions. The residue theorem is a useful
tool to calculate integrals associated with complex -
variable functions The residue theorem is an
important tool for the calculation of integrals of
complex variable functions.
In the course of researching complex variable
functions, for the calculation of the integrals of some
functions that have singular points in a closed region,
Cauchy's integral theorem cannot be directly applied
(Bak & Newman, 2010). However, the residue
theorem provides an effective method. It links the
residues of the function at each singular point within
the region enclosed by a closed curve with the integral
of the function along that closed curve. That is, the
integral of the function along the closed curve is equal
a
https://orcid.org/0009-0006-4658-8411
to 2π𝑖 times the sum of the residues of the function at
each singular point within the region enclosed by the
closed curve, which greatly simplifies the calculation
of the integrals of complex variable functions (Stein
& Shakarchi, 2003). The residue theorem deepens the
understanding of the singular points of complex
variable functions. The singular points of complex
variable functions are divided into types such as
removable singular points, poles, and essential
singular points. Through calculating the residues of
the function at singular points, the residue theorem
enables a more in-depth study of the properties and
characteristics of singular points. For example, by
calculating the residues, the author can determine the
type of a singular point and understand the local
properties of the function near the singular point (Chi,
2024).
The theoretical framework of complex variable
functions serves as the foundation for the residue
theorem. The analyticity of complex variable
functions, Cauchy's integral formula and other
theories are the basis for the derivation and proof of
the residue theorem. The properties of analytic
functions ensure the establishment of the residue
Xiao, Y.
Interdisciplinary Application of Residue Theorem in Trigonometric and Fraction Functions.
DOI: 10.5220/0013814900004708
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy (IAMPA 2025), pages 109-113
ISBN: 978-989-758-774-0
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.
109

theorem and provide various methods for calculating
residues, such as using the Laurent series expansion.
At the same time, some basic concepts and methods
of complex variable functions, such as contour
integrals and isolated singular points, are also the
prerequisites and keys for the application of the
residue theorem. The residue theorem promotes the
development and application of the theory of complex
variable functions. The emergence of the residue
theorem has promoted the extensive application of the
theory of complex variable functions in other fields.
It has played an important role in aspects such as
calculating real integrals, solving differential
equations, and studying fluid mechanics and
electromagnetics. These applications not only expand
the research scope of complex variable functions but
also further enrich and perfect the theoretical system
of complex variable functions.
2 METHODS
2.1 Complex Numbers and Functions
A complex number is of the form 𝑧=𝑥+𝑖𝑦 where x
and y are real numbers, and 𝑖 =
√
1
. A complex
function f(z) maps complex numbers to complex
numbers. One will review the basic operations on
complex numbers and functions, such as addition,
multiplication, and differentiation (Qiu, 2020).
For a function 𝑓(𝑧) that is analytic at an isolated
singularity z, its Laurent series expansion is
𝑓
(
𝑧
)
=𝑎

(𝑧 −𝑧

)



(
1
)
The residue is defined as the coefficient of
term
(
𝑧−𝑧

)

in the Laurent series, that is
Res
(
𝑓,𝑧

)
=𝑎

In practical calculations, for
different types of isolated singularities, there are
different methods for calculating the residue. If 𝑧

is
an mth order pole of 𝑓
(
𝑧
)
, then the formula
𝑅𝑒𝑠
(
𝑓,𝑧

)
=
𝑚
(
𝑚−1
)
!
lim
→

𝑓
(
𝑧
)(
2
)
can be used to calculate the residue. For example, for
the function𝑓
(
𝑧
)
=

(

)

(

)
, z = 1 is a second-
order pole and z = - 2 is a first-order pole. For z = 1,
according to the above formula, first let 𝑔
(
𝑧
)


,
then 𝑅𝑒𝑠
(
𝑓,𝑧

)
=


；for z=-2 𝑅𝑒𝑠
(
𝑓,𝑧

)
=−


.
Singularities are points where a complex function
is not analytic. Multiple kinds of singularities are
present, and among them are removable singularities,
poles, and essential singularities. The behavior of a
function near its singularities is crucial for
understanding the residue concept.
2.2 Residue Theorem
The residue theorem establishes a connection
between the integral of a function along a closed
curve and the residues of the function at the isolated
singularities inside the curve. Let f(z) be analytic in
the region D enclosed by a simple closed curve C
except for a finite number of isolated singularities
𝑧

𝑧

…，𝑧

, and continuous in the closed region 𝐷

=
𝐷∪𝐶 except at these singularities. Then one has

𝑓
(
𝑧
)
𝑑𝑧=2𝜋i𝑅𝑒𝑠(𝑓,𝑧

)



(
3
)
Below is the General Steps for Calculating Integrals
Using the Residue Theorem (Trefethen & Weideman,
2014).
The first is to determine the integration path.
Select an appropriate closed integration path C, which
is usually constructed according to the characteristics
of the integrand function and the integration interval.
For example, for the integral

𝑓
(
𝑥
)


𝑑𝑥on the real
axis, a semi-circular path (with radius R) in the upper
half-plane is often added, so that the integration path
C = 𝐶

+γ (γ is the line segment

−𝑅,𝑅

on the real
axis) forms a closed curve.
The second is to analyze the singularities of the
integrand function. Find all the isolated singularities
of the integrand function f(z) inside the integration
path C, and determine the types of the singularities
(removable singularities, poles, or essential
singularities). Then, according to the type of the
singularity, use the corresponding method to calculate
the residue at each singularity.
The third is to calculate the sum of the residues.
Add up the residues at all the singularities inside the
integration path C to obtain
∑
𝑅𝑒𝑠
(
𝑓,𝑧

)


.
The fourth is to apply the residue theorem to find
the value of the integral. According to the residue
theorem shown in Eq. (3), one can calculate the
numerical value of the integral taken along the closed
- loop curve C Then, by analyzing the limit situation
of the integral on the supplementary path (such as𝐶

)
when𝑅=∞, the value of the original integral on the
real axis can be obtained (Xu & Fan, 2024). For
example, when lim𝑅→∞
∮
𝑓
(
𝑧
)
𝑑𝑧=

2𝜋i
∑
𝑅𝑒𝑠(𝑓,𝑧



).
The residue theorem has many applications. In
physics, Cauchy's residue theorem is used in various
areas. For instance, in quantum field theory, it is used
to calculate scattering amplitudes and Green's
IAMPA 2025 - The International Conference on Innovations in Applied Mathematics, Physics, and Astronomy
110

functions. In statistical mechanics, it can be applied
to evaluate partition functions. The author will briefly
introduce some of these applications and explain how
the theorem is used in these contexts.
In Quantum Mechanics In quantum mechanics,
the residue theorem is used to calculate the Green's
functions. The Green's function is a powerful tool for
solving the Schrödinger equation and understanding
the propagation of quantum waves. For example, in
the study of scattering problems, the Green's function
can be expressed as a complex integral. By applying
the residue theorem, people can evaluate this integral
and obtain the scattering amplitude, which is a key
quantity in understanding the interaction between
particles.
In statistical physics, the partition function is a
central concept. It is often expressed as an integral
over a complex contour. The residue theorem can be
employed to evaluate this integral and obtain the
thermodynamic properties of the system, such as the
free energy, entropy, and specific heat. This allows
people to study the equilibrium and non - equilibrium
behaviours of physical systems.
3 APPPLICATION
3.1 Integrals of Trigonometric
Functions
Let 𝑧 = 𝑒

, then cos𝜃=
  


, sin𝜃=
  


, and 𝑑θ=𝑑𝑧 𝑖𝑧
⁄
(Hang et al, 2023). When 𝜃
varies from 0 to 2π, z makes a positive circuit along
the unit circle |z| = 1 in the complex plane. Thus, the
original integral

𝑅
(
cosθ,sinθ
)
𝑑


is transformed
into the contour integral of a complex function
I=  𝑅
𝑧 + 𝑧

2
,
𝑧 − 𝑧

2𝑖

𝑑𝑧
𝑖𝑧
|

|
 
(
4
)
Then, by using the Cauchy residue theorem, people
can calculate the residues of the integrand at the isol
ated singular points inside the unit circle |z| = 1, and
further obtain the value of the integral (Zeng, 2020).
For example, to calculate
I=
𝑑θ
1 + 𝑎cosθ
(|
𝑎
|
<1
)


(
5
)
Let 𝑧=𝑒

, the integrand is then transformed into
I=
2
(
𝑎 + 2
)
𝑧 + 𝑎𝑧

|

|
 
𝑑𝑧,
(
6
)
and in what follows one can introduce a new
function 𝑓
(
𝑧
)
=




(
  
)

. The singular points of
𝑓
(
𝑧
)
are 𝑧

= 0 and 𝑧

is not inside the unit circle.
𝑧

= 0 is a first-order pole, and Res

𝑓
(
𝑧
)
,0

=
lim
 → 
𝑧𝑓
(
𝑧
)
=

  
. According to residue theorem,

𝑑θ
1 + 𝑎cosθ


=
2π
√
1 − 𝑎

(
7
)
Next, the author will evaluate the integral

𝑓
(
𝑧
)
𝑑𝑧

.
(a) Evaluation of the integral for the semicircle z
= 2𝑒

(
0≤𝜃≤𝜋
)
.
First, the author shall express 𝑧 and d𝑧 in terms of
θ. Given z = 2𝑒

, then dz=2𝑖𝑒

𝑑𝜃. Substitute z into
the function f(z): 𝑓
(
𝑧
)
=


=1+


. Substituting z =
2e

, one gets 𝑓
(
𝑧
)
=1+



=1+𝑒

. Now, the
task is to calculate the contour integral

𝑓
(
𝑧
)
𝑑𝑧

, i.e.,

𝑓
(
𝑧
)
𝑑𝑧

=

1 + 𝑒




⋅2𝑖𝑒

𝑑θ. Expand the
integrand: 1 + 𝑒

⋅2𝑖𝑒

=2𝑖𝑒

+2𝑖 and
integrate it term - by – term, then
2𝑖𝑒

+2𝑖𝑑θ


=2𝑖𝑒



𝑑θ + 2𝑖 𝑑


θ
(
8
)
For

𝑒



𝑑θ, using the formula

𝑒

𝑑𝑥=


𝑒

+
𝐶
(
ℎ𝑒𝑟𝑒𝑎=𝑖
)
, one has

𝑒



𝑑θ=


𝑒




. Since


=−𝑖, then

𝑒



𝑑θ=−𝑖
(
𝑒

−𝑒

)
=−𝑖
(
−1 −
1
)
=2𝑖.

d𝜃


=𝜋.So, 2𝑖

𝑒



𝑑θ + 2𝑖

𝑑θ


=
2𝑖⋅ 2𝑖+ 2𝑖π=−4 + 2π𝑖.
(b) Evaluation of the integral for the semicircle
𝑧=2𝑒

(
𝜋≤𝜃≤2𝜋
)
.
Again, 𝑧=2𝑒

, 𝑑𝑧=2𝑖𝑒

𝑑𝜃, and 𝑓
(
𝑧
)
=1+


=1+𝑒

. Calculate the contour - integral

𝑓
(
𝑧
)
𝑑𝑧

:
𝑓
(
𝑧
)
𝑑𝑧

= 1+𝑒




⋅2𝑖𝑒

𝑑𝜃
(
9
)
Expand the integrand 1 +𝑒

⋅2𝑖𝑒

=2𝑖𝑒

+
2𝑖 and integrate term - by – term, then

2𝑖𝑒

+


2𝑖𝑑θ=2𝑖

𝑒



𝑑θ + 2𝑖

𝑑


θ.
For

𝑒



𝑑θ, using

𝑒

𝑑θ=−𝑖𝑒

+𝐶, one
has

𝑒



𝑑θ=−𝑖
(
𝑒
⋅
−𝑒

)
=−𝑖
(
1+1
)
=
−2𝑖 . So, it is calculated that 2𝑖

𝑒



𝑑θ +
2𝑖

𝑑θ


=2𝑖⋅
(
−2𝑖
)
+2𝑖π=4+2π𝑖.
(c) Evaluation of the integral for the circle 𝑧=
2𝑒

(
0≤θ≤2π
)
.
People can use the results from parts (a) and (b).
Since the circle 𝑧=2𝑒

(
0≤θ≤2π
)
is composed
of the two semicircles from parts (a) and (b), then
𝑓
(
𝑧
)
𝑑𝑧

=1+𝑒




⋅2𝑖𝑒

𝑑θ
Interdisciplinary Application of Residue Theorem in Trigonometric and Fraction Functions
111

+ 1+𝑒




⋅2𝑖𝑒

𝑑θ
(
10
)
From part (a), it is

1 + 𝑒




⋅2𝑖𝑒

𝑑θ=−4 +
2π𝑖,𝑎𝑛𝑑𝑓𝑟𝑜𝑚𝑝𝑎𝑟𝑡
(
𝑏
)
,

1 + 𝑒




⋅2𝑖𝑒

𝑑θ=
4+2π𝑖. Then it is found that

𝑓
(
𝑧
)
𝑑𝑧

=
(
−4 +
2π𝑖
)
+
(
4+2π𝑖
)
=4π𝑖.
3.2 Integrals of Fractional Function
The author shall define
𝑓
(
𝑡
)
=
𝑎𝑧

+𝑏𝑧

+𝑐𝑧+𝑑
𝑧

−1
(
11
)
with 𝑎=6,𝑏=𝑖+ 1,𝑐=16,𝑑=𝑖− 1 (Li et al,
2021). The task is to evaluate the integrals

𝑓
(
𝑧
)
d𝑧

with γ

(
𝑡
)
=𝑖+



,0≤𝑡≤2𝜋 γ

(
𝑡
)
=


+
√
2
𝑒

,0≤𝑡≤2𝜋 γ

(
𝑡
)
=1+5𝑒

,0≤𝑡≤2𝜋.
Firstly, people can factor the denominator 𝑧

−1.
People know that 𝑧

−1=
(
𝑧−1
)(
𝑧+1
)(
𝑧−
𝑖
)(
𝑧+𝑖
)
by the difference - of - powers formula
𝑎

−𝑏

=
(
𝑎−𝑏
)(
𝑎

+𝑎

𝑏+⋯+𝑎𝑏

+
𝑏

)
. Here, a = z, b = 1 and n = 4. So,
𝑓
(
𝑧
)
=
6𝑧

+
(
𝑖+1
)
𝑧

+ 16𝑧+
(
1−𝑖
)
(
𝑧−1
)(
𝑧+1
)(
𝑧−𝑖
)(
𝑧+𝑖
)
(
12
)
For the contour γ

(
𝑡
)
=𝑖+



,0≤𝑡≤2. The
center of the contour γ

is 𝑧

=𝑖 and the radius 𝑟=
12
⁄
. The singularities of 𝑓(𝑧) are the roots of 𝑧

−
1=0,i.e.,𝑧=1,𝑧=−1,𝑧=𝑖,𝑧=−𝑖.
People check which singularities lie inside the
contour γ

. The distance between a point z and the
center 𝑖 of the contour γ

is given by |z - i|. For 𝑧=
1,
|
1−𝑖
|
=

1+
(
−1
)

=
√
2>


. For 𝑧=
−1,
|
−1 − 𝑖
|
=

(
−1
)

+
(
−1
)

=
√
2>


. For
𝑧=𝑖,
|
𝑖−𝑖
|
=0<


. For 𝑧=−𝑖,
|
−𝑖 − 𝑖
|
=
|
−2𝑖
|
=2>


. By the residue theorem,

𝑓
(
𝑧
)
𝑑𝑧


=2π𝑖∙𝑅𝑒𝑠

𝑓
(
𝑧
)
. To find the residue at
𝑧 = 𝑖, people can use the formula 𝑅𝑒𝑠



(

)

(

)
=

(


)

′
(


)
where g
(
𝑧
)
=6𝑧

+
(
𝑖+1
)
𝑧

+ 16𝑧+
(
1−
𝑖
)
and ℎ
(
𝑧
)
=𝑧

−1.
Obviously, it is easy to find ℎ

(
𝑧
)
=4𝑧

. Then
ℎ

(
𝑖
)
=4𝑖

=−4𝑖. 𝑔
(
𝑖
)
=6𝑖

+
(
𝑖+1
)
𝑖

+ 16𝑖+
(
1−𝑖
)
=−6𝑖−
(
𝑖+1
)
+ 16𝑖+
(
1−𝑖
)
=
(
−6 −
1+16−1
)
𝑖+
(
−1 + 1
)
=8𝑖. So, 𝑅𝑒𝑠

𝑓
(
𝑧
)
=

(

)

′
(

)
=


=−2. Then

𝑓
(
𝑧
)
𝑑𝑧


=2π𝑖×
(
−2
)
=
−4π𝑖.
Secondly, for the contour γ

(
𝑡
)
=


+
√
2
𝑒

,0≤𝑡≤2, the center of the contour γ

is
𝑧

=


and the radius 𝑟=
√
2. Calculate the
distances from the singularities 𝑧=1,𝑧=−1,𝑧=
𝑖,𝑧= −𝑖 to the center𝑧

=


. |1 −


|=
√


=
√


>
√
2 . | − 1 −


|=
√


=
√


<
√
2 . |𝑖−


|=
√


=
√


<
√
2. | − 𝑖−


|=
√


=
√


>
√
2
. The singularities inside the contour γ

are 𝑧=
−1 and 𝑧 = 𝑖. By the residue theorem,
𝑓
(
𝑧
)
𝑑𝑧


=2π𝑖𝑅𝑒𝑠

𝑓
(
𝑧
)
+𝑅𝑒𝑠

𝑓
(
𝑧
)

(
13
)
For the residue at 𝑧=−1: ℎ

(
𝑧
)
=4𝑧

,ℎ

(
−1
)
=
−4 . In addition, 𝑔
(
−1
)
=6
(
−1
)

+
(
𝑖+
1
)(
−1
)

+16
(
−1
)
+
(
1−𝑖
)
=−6+
(
𝑖+1
)
−
16 +
(
1−𝑖
)
=−20. Thus, 𝑅𝑒𝑠

𝑓
(
𝑧
)
=

(

)

′
(

)
=


=5. Here, one has already found 𝑅𝑒𝑠

𝑓
(
𝑧
)
=−2.
So,

𝑓
(
𝑧
)
𝑑𝑧


=2π𝑖
(
5−2
)
=6π𝑖.
Thirdly, for the contour γ

(
𝑡
)
=1+5𝑒

,0≤
𝑡≤2, the center of the contour γ

is 𝑧

=1 and the
radius 𝑟 = 5. All the singularities 𝑧 = 1,𝑧=
−1,𝑧 = 𝑖,𝑧=−𝑖 lie inside the contour γ

. By the
residue theorem,

𝑓
(
𝑧
)
𝑑𝑧


=2π𝑖𝑅𝑒𝑠

𝑓
(
𝑧
)
+
𝑅𝑒𝑠

𝑓
(
𝑧
)
+𝑅𝑒𝑠

𝑓
(
𝑧
)
+𝑅𝑒𝑠

𝑓
(
𝑧
)
.
For the residue at z = 1:ℎ
′
(
𝑧
)
=4𝑧

,ℎ
′
(
1
)
=
4. 𝑔
(
1
)
=6×1

+
(
𝑖+1
)
×1

+16×1+
(
1−
𝑖
)
=6+
(
𝑖+1
)
+16+
(
1−𝑖
)
=24. 𝑅𝑒𝑠

𝑓
(
𝑧
)
=

(

)

′
(

)
=


=6. One found Res

𝑓
(
𝑧
)
=5,Res

𝑓
(
𝑧
)
=
−2 . For the residue at 𝑧=−𝑖:ℎ
′
(
𝑧
)
=
4𝑧

,ℎ
′
(
−𝑖
)
=4
(
−𝑖
)

=4𝑖 . 𝑔
(
−𝑖
)
=6
(
−𝑖
)

+
(
𝑖+1
)(
−𝑖
)

+16
(
−𝑖
)
+
(
1−𝑖
)
=6𝑖−
(
𝑖+1
)
−
16𝑖+
(
1−𝑖
)
=
(
−6−1−16−1
)
𝑖+
(
−1 + 1
)
=
−24𝑖. Hence, Res

𝑓
(
𝑧
)
=

(

)


(

)
=


=−6 .
Therefore,

𝑓
(
𝑧
)
𝑑𝑧


=2π𝑖
(
6+5−2−6
)
=
6π𝑖.
To conclude, for γ

, the integral is −4π𝑖; for γ

,
the integral is 6π𝑖; for γ

, the integral is 6π𝑖.
4 CONCLUSION
The residue theorem is of utmost importance in
mathematics, particularly in complex analysis. It
serves as a powerful tool for evaluating complex
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112

integrals, providing an efficient method to calculate
the values of integrals that would otherwise be
extremely difficult or even impossible to solve using
traditional real - variable methods. For numerical
algorithms with the development of computer
technology, more efficient numerical algorithms
based on the residue theorem will be developed.
These algorithms will be able to handle large - scale
and high - dimensional integral problems, providing
powerful computational tools for both theoretical and
applied mathematics. It can cross disciplinary
applications and for differential Equations: There will
be more in - depth connections with complex - valued
differential equations. The residue theorem can assist
in solving certain types of differential equations by
transforming them into integral problems and then
using residue - based methods for solution. For
physics the residue theorem applies in many fields
such as Quantum Mechanics statistical physics, and
the residue theorem is an indispensable tool in
physics. Its applications in quantum mechanics,
electromagnetism, and statistical physics have
enabled people to solve complex problems and gain a
deeper understanding of physical phenomena. As
physics continues to advance, the residue theorem
will likely find even more applications in new and
emerging areas, further enhancing people’s ability to
describe and predict the behaviour of the physical
world.
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Hang, Z. X., Du, X. R., & Ma, Y. L. (2023). Generalization
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