Applications of Cauchy's Residue Theorem in Complex Functions of
Fractional Form
Wenshuo Li
a
Qingdao NO.58 middle School, Qingdao, China
Keywords: Cauchy’s Residue Theorem, Singularities, Laurent Series, Taylor Series.
Abstract: This paper explores the applications of Cauchy's Residue Theorem in complex analysis, focusing on its utility
in evaluating complex integrals around closed contours. The study begins with an introduction to the Laurent
series and Taylor series, which are foundational for understanding the Residue Theorem. The Residue
Theorem uses these series to find "residues", which is a fancy term for coefficients that capture the behavior
at singularities. Summing these residues gives the integral’s value instantly. The paper then delves into the
theorem's theoretical framework, illustrating its application through several examples, including functions
with simple poles, higher-order poles, and removable singularities. For simple poles (basic singularities),
calculating residues is straightforward. For harder cases (like higher-order poles), it needs the use of
derivatives. The results demonstrate the theorem's effectiveness in simplifying complex integral calculations,
particularly in cases involving trigonometric and rational functions. The research highlights the theorem's
significance in both theoretical mathematics and practical applications, such as physics and engineering. The
paper concludes with a discussion on the potential for further exploration and the implications of these
findings for advanced mathematical studies.
1 INTRODUCTION
Functions of complex variables are the primary
subject of study in complex analysis, a mathematical
branch (Ahlfors, 1979). One of the most powerful
tools in this field is Cauchy theorems, which enables
a precise calculation for integrating by summing the
coefficients obtained through Laurent series
expansions at critical points. This theorem is
particularly useful in solving real integrals that are
otherwise difficult to compute using standard
techniques. Among its key results, Cauchy's Residue
Theorem stands out as a "mathematical supertool"
that transforms intricate integrals into simple
algebraic computations by leveraging singularities,
representing points where functions behave
abnormally (Shen & Li, 2016). This theorem bridges
pure theory and applied mathematics, offering unified
solutions to problems ranging from electromagnetic
field calculations to signal processing.
The importance of Cauchy's Residue Theorem
extends beyond pure mathematics. It has significant
applications in physics, engineering, and other
a
https://orcid.org/0009-0009-5663-7207
sciences where complex integrals frequently arise. In
physics, it provides exact solutions to problems in
fluid dynamics and quantum mechanics. Engineers
rely on it for signal processing algorithms and control
system design. Even in pure mathematics, it aids in
number theory through the study of zeta functions.
The Residue Theorem's significance is well-
documented in both historical and modern contexts
(Shen, 2017). Ahlfors noted its role in 19th-century
function theory development, while contemporary
researchers like Stein & Shakarchi emphasize its
utility in evaluating Fourier and Laplace transforms.
In physics, Peskin et al demonstrate how residues
simplify Feynman path integrals for particle
interactions. In this essay, Cauchy’s Residue
Theorem could be extended to solve higher order
singularities.
Section breakdown is organized in this
arrangement. Section 2 introduces the Laurent series
and Taylor series, which are essential for
understanding the Residue Theorem. Section 3
presents several examples demonstrating the
theorem's application, including functions with
Li, W.
Applications of Cauchy’s Residue Theorem in Complex Functions of Fractional Form.
DOI: 10.5220/0013814700004708
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 99-103
ISBN: 978-989-758-774-0
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.
99
simple poles, higher-order poles, and removable
singularities. This paper addresses both gaps by
presenting visualizable examples and systematic
methods for handling second-order pole. There are
examples in Section 3 showing how to apply the
theorem to second order poles. Final section
synthesizes key discoveries and present forward-
looking propositions.
2 METHOD AND THEORY
2.1 Laurent Series & Taylor Series
Before getting into Cauchy’s Theorem, it is necessary
to mention the two series that it is based on.
The mathematical apparatus of Laurent series
constitutes a generalization of Taylor series to
accommodate singularities in complex function
representation (Zhou, 2022). While a Taylor series
can only represent functions exhibiting differentiable
behavior within a neighborhood of dot, Laurent series
can represent functions that have singularities, such
as poles or essential singularities, within the region of
interest. This gives Laurent expansion unique
advantages in handling functions with singularities.
The Laurent series of a holomorphic expression
Γ
(
τ
)
within annulus of τ
shows:
Γ
(
τ
)
=α
(
τ−τ
)
+β
(
τ−τ
)
(
1
)
where α
are complex-valued coefficients, and the
series expansion involves negative coefficients. Its
convergence holds in an annular region,
mathematically characterized as the open set between
two circles with identical centers but distinct radii.
Geometrically, an annulus refers to the ring-shaped
domain encircled by an inner circumference with a
radius of r
and an outer circumference with a radius
of r
, both centered at the same point.
Laurent’s formulation/expansion can be split into
two parts, the principal part and the analytic part. This
principal series representation includes a component
featuring inverse powers of (z − z
) , specifically the
summation from n=−∞to n=0 of all powers.
This part captures the function's behavior in the
vicinity of the singularity z
is of particular interest.
Containing non-negative terms of (z − z
), the series
are summation of all powers where n ranges over all
natural numbers starting from zero. This part behaves
like a Taylor series and represents the analytic part of
the function, the coefficients are determined by the
following formula, thereby facilitating accurate
calculations.
α
=
1
2πi
Γ
(
ξ
)
(
ξ−z
)
dξ
(
n=0,±1,±2
)
.
(
2
)
Of note is that Laurent series still works if z
is an
isolated singularity. The residue of the function at the
specified point is precisely determined by the
corresponding coefficient.
By making Γ
(
τ
)
differentiable on
(
z−z
)
< R,
then all b
= b
= b
⋯⋯ b
=0, the Laurent
series is then weakened (reduced) to become a Taylor
series. For example, people can focus on the Laurent
expansion, Γ
(
τ
)
=
within annulus of τ=0. For
|τ|<2, it is found that Γ
(
τ
)
=
∑
, but for
|τ|>2, Γ
(
τ
)
=
∑
. As a special case, when
The elimination of all principal part terms transforms
the Laurent expansion turns into a Taylor series,
which states if Γ
(
τ
)
exhibits the property of complex-
differentiability in |z − z
|<R, then for any 𝑧 in that
region, one has that Γ
(
τ
)
=
∑
α
(
τ−τ
)
, and
the coefficient in this power series given by a
=
()
(
)
!
(
n=0,1,2,3,⋯n
)
. Especially, a
=f
(
z
)
,
a
= f
(
τ
)
1
⁄
, a
= f"
(
τ
)
2
⁄
, and so on. It is a
series like a polynomial but of infinite degree.
2.2 Cauchy's Theorem
This paper starts by talking about Cauchy-Goursat
Theorem, which states that if the function 𝑓(𝑧)
exhibits differentiability at each point within the
interior of a simple closed curve 𝐶, then
f
(
z
)
dz = 0
(
3
)
The theorem reveals that the line integral of an
analytic complex function around any closed contour
always equals zero (Lin, 2021). This is a pivotal
finding in complex analysis, demonstrating that if a
function is holomorphic within a simply connected
region, its line integral becomes independent of the
specific integration path chosen. Second, about
Cauchy's renowned contour integral relation, which
states that if given a closed curve with f(z) being
analytic in its entirety, then
f
(
z
)
=
f
(
z
)
z−z
dz
(
4
)
This theorem is derived from the Cauchy-Goursat
Theorem (Mitrinović & Kečkić, 1984). It reveals that
the integral of an analytic function along a closed
contour can be represented using the values of the
function within the contour. This formula is
particularly useful for calculating the values of
analytic functions. Finally, all above leads to the
Cauchy’s Theorem. Suppose 𝐶 is a simple closed
IAMPA 2025 - The International Conference on Innovations in Applied Mathematics, Physics, and Astronomy
100
contour, and 𝑓(𝑧) is analytic at all points within C,
with the exception of the singularities
(
z
,z
,z
⋯z
)
then,
Γ
(
τ
)
dτ
= 2πiRes
(
Γ,τ
)
+Res
(
Γ,τ
)
+⋯+Res
(
Γ,τ
)(
5
)
Here, Res
(
Γ,τ
)
denotes the residue of Γ
(
τ
)
at the
singularity z
. The residue can be regarded as the
coefficient of 1
(
z−z
)⁄
in the Laurent expansion.
The Residue Theorem serves as a pivotal
instrument within the realm of complex function
theory. It offers an approach to evaluate contour
integrals in the complex plane, especially when
dealing with functions that possess isolated
singularities. This theorem enables the computation
of integrals of complex functions along closed
contours by aggregating the residues of the function
at its singular points enclosed by the contour.
In the context of complex analysis, the residue of
a complex function 𝑓(𝑧) at an isolated singularity z
can be identified as the coefficient associated with the
term within the Laurent series expansion of 𝑓(𝑧)
centered at z
. The calculation of residues typically
depends on the type of singularity. For simple poles,
the residue can be directly computed using limits,
while for higher-order poles, a combination of
derivatives and limits is required.
The connections of three theorems are the
following. Initially, the Cauchy-Goursat Theorem
serves as the cornerstone, demonstrating that the
integral of an analytic function along a closed contour
equals zero. This fundamental principle underpins the
Cauchy Integral Formula. Subsequently, the Cauchy
Integral Formula acts as an expansion of the Cauchy-
Goursat Theorem, enabling the computation of the
value of an analytic function within a given path
through the application of an integral. Third, the
Theorem further extends the Cauchy Integral
Formula, enabling people to solve integrals over
closed paths by computing the residues of the
function at its singular points. When a function has
singularities inside the path, the Cauchy-Goursat
Theorem no longer applies, but the Cauchy’s Residue
Theorem remains valid.
3 RESULTS AND APPLICATIONS
In the following examples, there would be three main
categories. The first category is about simple
fractions; the second is about fractions which both
numerators and denominators are complex; while the
third is about function with removable singularity and
second poles (Xu & Fan, 2024).
3.1 Example Application 1
In this section, Residue Theorem is applied to simple
fractions in which only the denominator is complex
while the numerator is rational number (He, 2021).
The initial approach involves examining the
function f
(
z
)
=
. It has simple poles at z=i
,z=−i. To compute the integral about a contour C=
1 + 4e
that contains these poles, one first finds the
residues at these poles. For the pole at z=i
:Res
(
f,i
)
=lim
→
(
z−i
)
=
. For the pole at z=
−i : Res
(
f,−i
)
=lim
→
(
z+i
)
=
. Then by
applying Cauchy's Theorem, it is found that
1
z
+1
dz
=2πi
1
2i
+
1
−2i
=0
(
6
)
This result is consistent with the fact that f(z) is
analytic all points except for z=i,z=−i, and the
residues at these points cancel each other out.
The second is to consider function Γ
(
τ
)
=
.
It has simple poles at τ=1 ,τ=−1. To compute the
integral about a contour C= 1+5e
that encloses
these poles, first find the residues at each pole (Lin &
Gong, 2018). For the pole at τ=1: Res
(
Γ,1
)
=
lim
→
(
τ−1
)
=
. For the pole at τ=
1 : Res
(
Γ,−1
)
=lim
→
(
τ+1
)
=−
. The
integral is found to be
∮
dτ
=0. This result is in
line with the fact thatΓ
(
τ
)
is analytic except for two
points at which the residues cancel each other out.
One can also consider the function f
(
z
)
=
(
)
.
This function has simple poles at z=1. To compute
the integral of around a contour C= 1+6e
that
encloses these poles, first find the residues at the pole.
For the pole at z= 1: Res
(
f,1
)
=lim
→
(
z−
1
)
(
)
=1. Then by applying Cauchy's Residue
Theorem, the integral is:
1
(
z−1
)
dz=2πi
(
7
)
This result is consistent with the fact that is analytic
everywhere except at, and the residues at these points
cancel each other out.
3.2 Example Application 2
Cauchy’s Residue Theorem is applied to fractions
with both numerators and denominators be complex.
Applications of Cauchy’s Residue Theorem in Complex Functions of Fractional Form
101
The author shall focus on the function f
(
z
)
=
z
z
+1
⁄
. This function has singularities at z=
i,z=−i. Then the author will calculate the residues
at two singularities. The residues of a function f
(
z
)
at
a first-order pole z=a is given by Res
(
f,a
)
=
lim
→
(
z−a
)
f
(
z
)
. Thus, it is calculated that Res
(
f,i
)
=
lim
→
(
z−i
)
(
)(
)
=
as well as Res
(
f,−i
)
=
lim
→
(
z+i
)
(
)(
)
=−
. The final integral is
z
1+z
= 2πi
e
⁄
2i
−
e
⁄
2i
=πe
⁄
−e
⁄
(
8
)
Another function is like this f
(
z
)
=
(
)(
)
. It
has first-order poles at z=−1 ,z=
. To compute
the integral about contour C= 3+4e
that contains
these poles, first find the residues at each pole. It is
found that Res
(
f,−1
)
=2 and Res
f,
= 2 3
⁄
.
The final integral could be calculated by using
Cauchy’s Residue Theorem
2z
−z+1
(
2z − 1
)(
z+1
)
dz=2πi2 +
2
3
=
16πi
3
(
9
)
Finally, the author will consider a more complex
function (with trigonometry functions) like this:
f
(
z
)
=
(
)
. To calculate the integral associated
with contour C = i + 5e
, one can first calculate the
three residues at z=0,z=i,z=−i. It is found that
Res
(
f,i
)
=
, Res(f,−i) =
, Res
(
f,0
)
=
=0. Thus, using the Cauchy’s Residue
Theorem, the integral is
f
(
z
)
dz
=2πi
(
0+1−cosi
)
=2πi
(
1 − cosi
)(
10
)
3.3 Enhanced Application
In this section, Residue Theorem is applied to a
function with removable singularity and second
poles.
Consider the function like this
f
(
z
)
=
z
(
z
+π
)
sinz
.
(
11
)
To calculate the integral associated with contour
C = i + 6e
, it is found that it has one pole at z=
0 , and Res(f,0)= lim
→
(
)lim
→
(
(
)
)=0
(Labora & Labora, 2025). It is inferred that z=iπ is
a pole of order 2. Res
(
f,iπ
)
=−
+
.
Thus, it is calculated that
f
(
z
)
dz=
i
2
−
1
sinhπ
+
coshπ
sinhπ
.
(
12
)
4 CONCLUSIONS
To summarize, this paper has explored the
applications of Residue Theorem in complex analysis
and has stated its usefulness in calculating complex
integrals around closed contours. Through various
examples, how the theorem simplifies the calculation
of integrals involving functions with singularities is
clearly shown. The results highlight the theorem's
utility in both theoretical and applied contexts,
including trigonometric function, fractional function,
and others. Future research could explore the
theorem's applications in more complex scenarios,
such as functions with essential singularities or in
higher-dimensional spaces. Additionally, further
investigation into the computational aspects of the
theorem could lead to more efficient algorithms for
solving complex integrals. Overall, Cauchy's Residue
Theorem remains a cornerstone of complex analysis,
with wide-ranging implications for evaluating
integrals with higher order poles.
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Lin, J., & Gong, M. (2018). Application of residues in
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