Complex Analysis Approach to Solving Problems with Trigonometric
Functions
Wenxuan Zhang
a
Shandong Weifang Wenhua Senior High School, Weifang, China
Keywords: Residue Theorem, Trigonometric Functions, Complex Function, Definite Integral.
Abstract: Nowadays, many problems are involved with the integrals. Whether in the field of physics or mathematics, if
people want to solve some intricate integration, there is no doubt that one should use the residue theorem.
This paper mainly focuses on how to solve some complex integration by using the residue theorem. By
enumerating some examples about the integrations in complex plane, the author shall show the effects on
different field to prove the prominence of residue theorem. By this paper, it will embody and learn to the
greatest extent possible. The meaning of this paper is that it shows the importance of residue theorem, and
give person another tool to solve some involute integrations in complex plane. What is more, this paper can
show the importance of residue, which not only can apply in mathematics, but also can use in considerable
fields. In addition, it can help people analyse the earthquake, electricity and a huge variety of difficult
integrations.
1 INTRODUCTION
In the era of rapid development, a huge variety of
integration should be solved, regardless of in
mathematics or physics, even in geography. In order
to solve these difficult problems, one must use residue
to reduce the difficulty of integrations. For example,
in multi-dimensional and multi-point earthquake
response spectrum, there are considerable problems
such as lower rate of calculation and this deviates
from the original simplicity and high efficiency of the
response spectrum method and. Analyzing multi-
dimensional and multi-point earthquake response
spectrum and based on wavelet transform, a seismic
simulation algorithm is proposed. This method can fit
the standard response spectrum, but does not fully
consider the spatial effect factors. This method also
has some problems in duration, but applying the
residue can reduce the time of calculation to a certain
degree (Zhao et al, 2022).
In addition, the residues not only help researcher
analyze the earthquake, but also can help investigator
finish the calculation and analysis of flexible DC
transmission system (Sakhaei, 2025). Use the residue
theorem to calculate the αβ component vector of the
AC voltage on the complex plane and estimate the
a
https://orcid.org/0009-0008-8066-3547
angular frequency (Kaitlin & Patricia, 2022). The
residue theorem can be used to calculate the integral
of the complex function 𝑓𝑧 along a closed path 𝐶
containing several singular points. Comparing with
the conventional method, it can provide higher
efficiency. The complex function 𝑓𝑧 can be
designed according to the expected interference and
noise suppression characteristics of the frequency and
phase angle estimator. What is more, it also can
simplify the formula for the path integral and the
control structure of the estimator is designed (Liu,
2023). In Physics when researching for topological
phase, it can judge a phase is either mediocre or
topological in a system by depending on topological
invariants in a system. However, when calculating the
topological invariants, it needs to definite the
transformation of Hamiltonian from lattice space to
momentum space, thus in this process, it must use the
residue theorem to deal with this problem. It can
transform the loop integration problem of an analytic
function into the problem of finding the sum of the
residues of the singular points of the integrand (Meng
& Guan, 2023).
In this paper, the author will describe the
definition of the residue theorem and its formular at
first. At the same time, the work will give some
94
Zhang, W.
Complex Analysis Approach to Solving Problems with Trigonometric Functions.
DOI: 10.5220/0013814600004708
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 94-98
ISBN: 978-989-758-774-0
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.
examples about some problems which can transform
to use the residue theorem to make it easier to
accomplish. It will also tell the applications about
residue theorem in variance field, mainly showing
how to use the residue theorem in trigonal functions.
Finally, it shows a conclusion of the residue theorem
with some meaningful value and its future
development and use.
2 METHODS
2.1 Residue Theorem
In complex analysis, Residue Theorem is a
paramount tool to exploit the path integral of an
analytic function along a closed curve. What is the
meaning of Residue? The meaning of it is that it refers
to the integral of an analytic function along any
positive simple closed curve surrounding an isolated
singular point in a circular domain divided by 2𝜋𝑖.
The formula of this theorem is given by
𝑓
𝑧
𝑑𝑧
=2π𝑖𝑅𝑒𝑠
𝑓
𝑏
1
In this formula, the part of Res
𝑓
𝑏
means that
the residue of 𝑓
𝑧
is in 𝑏
and it is identical with the
coefficient，𝑎
，in
𝑧−𝑏
Laurent Expansion
in neighbourhood of 𝑏
. The Laurent expansion plays
an essential role in complex function, due to the fact
that it is a form of series in isolated singularity. It can
give the value of residue directly, so that people can
use it in some specific condition rather than the
residue theorem (Wu, 2011).
When it comes to the Laurent expansion, it must
think about the singularity (Labora & Labora, 2025).
The singularity can be divided into many categories,
but in residue field, the essential singularity and
removable singularity usually be used in a wide range
of problems. Especially removable singularity, the
residue of it is zero. In addition, the pole is another
paramount factor. For example, the first-order pole in
Laurent expansion, the maximum negative power
term is
𝑧−𝑧
. Otherwise, the essential
singularity has infinitely many negative powers. Such
as z=0 is an essential singularity of f
z
=𝑒
.
When calculating the residue, one firstly needs to
identify the types of singularity. For different types of
singularity, different methods can be used. The
application of Residue theorem is in many respects.
For instance, Rational fraction expansion and
trigonometric functions are nee to use residue
theorem.
2.2 Applications
The residue theorem is not only used in mathematics,
but also can be applied in physics. For instance, in
Quantum Mechanics, the residue theorem probably
facility to calculate of integration in Green’s function,
especially in energy level calculation (Lin, 2015).
What is more, it can also determine location of energy
levels and resonance states in a system rapidly by
using residue theorem. Otherwise, it will clarify the
integration of oscillating function in electromagnetic
fields. In addition, in statistical mechanics, it is a
predominant way to solve some complex integration,
specifically in Gaussian Integral and Fourier
Transform. In addition, considerable critical
phenomenon problems can be solved by utilizing the
residue theorem. In the field of signal processing, the
residue is the priority due to the fact that it could make
it easy to a certain extent. For example, in filter
frequency response, it can transform the integration
in complex plane.
2.3 Trigonometric Functions
The trigonometric functions are essential too. It can
help people to solve some problems in triangle. The
sine, cosine and tangent are always used. They have
some equations. For example, sin
α + cos
α=1,
sin2𝛼 =2sin𝛼 cos𝛼, cos2α =cos
α−sin
α=
1−2sin
α (Liu, 2019). These equations can help
people to transfer the difficult function to easy one.
Because it is a function, it also has the same properties
of function. For instance, they have the cycle, but they
are not same. The tangent function is π. The sine and
cosine are 2π. Otherwise, the parity is not same too.
The parity of sine and tangent are odd function, and
the cosine function is even function.
The Odd/Even Symmetry can help people
simplify the calculations. In addition, the sine and
cosine functions are Bounded Functions, nevertheless
the tangent is not bounded functions. What is more,
symmetry is different too. For example, the sine
function is symmetric about the origin, and the cosine
function is symmetric about the y-axis, due to even
function. The tangent function is specific one. It is
symmetric about the origin, and the image is
symmetric about each
+kπ point.
3 EXAMPLES
3.1 Basic Setups
The first example is to evaluate the integral
Complex Analysis Approach to Solving Problems with Trigonometric Functions
95
I=
𝑧+1
𝑧
𝑑𝑧
|
|
2
The author will use both direct contour integration
method and also the calculus of residues to solve this
integral. In this first method, one can parameter the
integration in a unit circle with
|
𝑧
|
=1. Depending
on Euler's formula, 𝑧= 𝜌𝑒
, it can transform to 𝑧=
𝑒
. The angle 𝜃 is from 0 to 2π, thus 𝑑𝑧= 𝑖𝑒
𝑑𝜃.
So, the integration can transform to
I=
𝑧+1
𝑧
dz
|
|
=
𝑒
+1
𝑒
∙𝑖𝑒
dθ
=i 𝑒
+1dθ
3
The author calculates respectively
𝑒
𝑑𝜃 =
0 and
1𝑑𝜃 =2𝜋𝑖
. Therefore,
𝐼=𝑖∙
2𝜋+ 0
=2𝜋𝑖.
4
In the second method, there is a second-order pole in
𝑧=0 of an integrand 𝑓
𝑧
=
. So, one can
calculate the residue as
Res
𝑓,0
=lim
→
𝑑
𝑑𝑧
𝑧
∙
𝑧+1
𝑧
=lim
→
𝑑
𝑑𝑧
𝑧+1
=1
5
Depending the residue theorem, one can calculate the
points as
2𝜋𝑖∙ 𝑅𝑒𝑠
𝑓,0
=2𝜋𝑖∙1=2𝜋𝑖
6
The residue theorem not only can solve some
integration in complex variable function, but also can
complete some real integral especially in
trigonometric functions. In a study of real integral
which is about trigonometric functions, it has a
common form:
𝐼= 𝑅
cos𝜃,sin𝜃
𝑑𝜃
.
7
The definition of 𝑅
cos𝜃,sin𝜃
is a rational
function in a unit circle with a range of
|
𝑧
|
=1. There
are considerable replacements to make real integrals
be transformed into closed integrals around the unit
circle. Let 𝑧=𝑒
, then cos𝜃 =
, sin𝜃 =
, 𝑑𝜃=
. The initial integration can be
transformed:
𝑅
cosθ,sinθ
𝑑θ= 𝑓
𝑧
𝑑𝑧
𝑖𝑧
|
|
8
3.2 Examples
The author now calculates this integration
I=
cosθ
5 + 4cosθ
𝑑θ
9
Let 𝑧=𝑒
,cos𝜃 =
,𝑑𝜃 =
, and the path of
integration in complex plane is a unit circle with
|
𝑧
|
=1. The initial integration has changed to
∮
∙
∙
|
|
, and simplify denominator and
numerator:
z
+1
2iz ∙
2z
+5z+2
dz
|
|
10
The roots of 2𝑧
+5𝑧+2 are respectively 𝑧=
−
and 𝑧=−2. However, in a unit circle with
|
𝑧
|
=
1 only have 𝑧=0 and 𝑧=−
to be poles. When z is
equal to 0 (it is a first order pole):
Res
𝑓,0
=lim
→
𝑧∙
𝑧
+1
2𝑖𝑧 ∙
2𝑧
+5𝑧+2
=
1
4𝑖
11
When z is equal to −
, then Res
𝑓,−
=
lim
→
𝑧+
∙
∙
=−
. By using the
residue theorem, the total residue is:
−
=−
(Xu & Fan, 2024). The result of integration is: 2𝜋𝑖∙
−
=−
. However, people should test the result,
and depending on standard integration formula one
can get:
𝐼=
cos𝜃
5 + 4cos𝜃
𝑑𝜃
=−
𝜋
3
12
Next, the author calculates the integration:
I=
𝑑θ
1 + 𝑎cosθ
|
𝑎
|
<1
13
Let 𝑧=𝑒
and author uses some replacement to
transform the initial integration. The replacement is
cos𝜃 =
, 𝑑𝜃 =
and the unit circle of
|
𝑧
|
=
1. The final integration is
I=
1
1+𝑎∙
𝑧+𝑧
2
∙
𝑑𝑧
𝑖𝑧
|
|
=
2
𝑎𝑧
+2𝑧+𝑎
∙
𝑑𝑧
𝑖
|
|
14
The author now analyses the singularity. The root
of 𝑎𝑧
+2𝑧+𝑎=0 is 𝑧=
±
but only 𝑧
=
is in this unit circle, due to the fact that
|
𝑎
|
<1,and test the length of 𝑧
,
|
𝑧
|
<1. The
integrand is:𝑓
𝑧
=
. The residue in the
pole 𝑧
is given by Res
𝑓,𝑧
=lim
→
𝑧−𝑧
∙
IAMPA 2025 - The International Conference on Innovations in Applied Mathematics, Physics, and Astronomy
96
=
. Taking 𝑧
−𝑧
=
into the residue, it is found that Res
𝑓,𝑧
=
.
By using the residue theorem, it is found that
2πi ∙ 𝑅es
𝑓,𝑧
=2πi∙
1
𝑖
√
1−𝑎
=
2π
√
1−𝑎
15
The author also evaluates the integral
I=
1
𝑎 − 𝑏 cosθ
𝑑θ,𝑎>𝑏>0
16
One can use residue theorem to transform to complex
integral. The author expands the range of integral
from 0 to 2π.
Let 𝑧=𝑒
, so it has changed to cos𝜃 =
and 𝑑𝜃 =
. The initial integral transforms to: I=
𝑑𝑧
|
|
. The roots of −𝑏𝑧
+2𝑎𝑧−𝑏
are 𝑧
=
+
−1 and 𝑧
=
−
−1 .
𝑧
is in this unit circle due to 𝑎>𝑏>0. The author
calculates the residue in 𝑧
:
Res
,𝑧
=−
. So 𝐼=
.
The author calculates the I=
∮
𝑑𝑧
|
|
.
People can divide the
𝑧
+1
into
𝑧−𝑖
𝑧+
𝑖
. So, there are two double poles, z=i and z=−i.
When z=i, Res
f,i
=−
=
. When z=−i,
Res
f,−i
=−
=−
. Thus, the author will
calculate the sum of these two residues, and it is equal
to zero.
The author will calculate integral I=
.
Let 𝑧=𝑒
, so dθ=
, cosθ=
,
cos2θ=
. The integration will transform to
I=
𝑧
+𝑧
2
5−2
𝑧+𝑧
∙
𝑑𝑧
𝑖𝑧
|
|
=−
1
2𝑖
𝑧
+1
𝑧
2𝑧 −1
𝑧−2
𝑑𝑧
|
|
17
There are two poles, respectively, z=0 and z =0.5. The
author will respectively calculate the residue. When
z=0, Res
f,0
=−
. When z=0.5, Res
f,0.5
=
.
So, the total residue is
. Then, the author will use
the residue theorem to calculate the integration:
I=2𝜋i∙
1
12𝑖
=
π
6
18
The author will show the detailed process of this
integration:
𝐼=
𝑥cos𝑥
𝑥
−7𝑥+10
𝑑𝑥
19
By dividing the denominator 𝑥
−7𝑥+10 into
x−
2
x−5
, the initial formula can be expressed into
practical fractions:
−
. The author transforms
the initial integration into: I=
−
∞
∞
𝑑𝑥
∞
∞
. Using complex integrals and residue
theorem, it is found that P.V.
𝑑𝑥 =
∞
∞
−πsin𝑎 . When 𝑎=5 and 𝑎=2,
𝑑𝑥 =
∞
∞
−πsin5,
𝑑𝑥 = −πsin2
∞
∞
. So,
I=
5
3
−πsin5
−
2
3
−πsin2
=
π
3
2sin2− 5 sin5
.
20
4 CONCLUSIONS
All in all, the residue theorem can provide people
another idea to exploit the path integral of an analytic
function along a closed curve. It can help researcher
to solve the complex integration with a higher
efficiency. In this paper, the author wants to show
more higher efficiency method to solve a huge variety
of mathematics. To this end, the author also hopes
that there would be more and more new and easier
method instead the conventional method. In this
paper, it nevertheless has some defects. For instance,
the example is easier than the real-world problems,
due to the fact that it only depends on theory to solve
the troubles in the mathematics instead of adding real
conditions. It may have a little difference with the
real-world problems. In the future, the author wants
to study more theory about the residue to provide
more and more convenient method to solve these
difficult integrals. In addition, the author also hopes
to study some real-world problem to realize the ideas
of combining theory with practice.
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