American Journal of Applied Science and Technology
26
https://theusajournals.com/index.php/ajast
VOLUME
Vol.05 Issue01 2025
PAGE NO.
26-29
10.37547/ajast/Volume05Issue01-07
Bessel functions of the first kind
Raupova Mohinur Haydar qizi
Chirchiq State Pedagogical University, Uzbekistan
Xasanova Mohichexra Farxod qizi
3rd year student, Chirchiq State Pedagogical University, Uzbekistan
Received:
25 October 2024;
Accepted:
28 December 2024;
Published:
30 January 2025
Abstract:
This paper discusses the derivation of Bessel functions of the first kind using power series method and
their properties. Additionally, the practical applications of these functions, their graphical analysis, and
relationships with other special functions are examined. The research results serve to expand the theoretical and
practical significance of Bessel functions.
Keywords:
Bessel functions, power series, mathematical analysis, physical phenomena, engineering problems,
special functions.
Introduction:
Bessel functions are widely used in
problems with cylindrical symmetry, particularly in the
analysis
of
vibrations,
heat
transfer,
and
electromagnetic waves. These functions have found
their significant place in numerous physical
phenomena and engineering problems, especially in
the analysis of oscillations, heat conduction, and
electromagnetic wave propagation. Moreover, their
graphical
analysis
and
application
through
mathematical formulas provide more effective
solutions to physical problems.
The paper illustrates the derivation of Bessel functions
of the first kind using power series method and related
mathematical properties. Furthermore, it analyzes
these functions' applications in physical problems, their
practical significance, and relationships with other
special functions. The research results contribute to the
broader application of Bessel functions and expand
their importance in scientific research, engineering,
and physics.
Bessel functions are typically defined through
differential equations and are derived using power
series. The Bessel function of the first kind is written as:
( )
2
0
1
2
( )
2
!
(
1)
n v
n
n
n
n
x
x
J x
n Г v
n
+
=
−
=
+ +
where
( )
n
J
x
−
is the Bessel function,
x
−
is the argument,
v
−
is the order index, and
Г
−
is the Gamma
function.
The derivation of Bessel functions is accomplished through the Bessel equation:
(
)
2
''
'
2
2
0
x y
xy
x
v
y
+
+
−
=
(1)
Or
American Journal of Applied Science and Technology
27
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
'
2
''
1
0
y
v
y
y
x
x
+
+ −
=
(2)
this equation is called the Bessel equation, where the constant
v
−
is called the index of the equation.
In the next step, we introduce the substitution
0
v
into the equation
v
y
x z
=
.
(
)
(
)
'
1
'
1
2
1
'
'
''
'
(
1)
2
'
''
v
v
v
v
v
v
v
v
y
x
z
v x
z
x
z
y
v x
z
x
z
v v
x
z
v x
z
x
z
−
−
−
−
=
=
+
=
+
= −
+
+
'
2
''
1
0
y
v
y
y
x
x
+
+ −
=
1
2
2
1
'
(
1)
2
'
''
1
0
v
v
v
v
v
v
vx
z
x
z
v
v v
x
z
v x
z
x
z
x z
x
x
−
−
−
+
−
+
+
+
+ −
=
Simplifying
this expression and to simplify the function, we obtain the equation
z
:
''
(2
1)
'
0
v
z
z
z
x
+
+
+ =
(2)
We seek the solution of this equation in the form of:
0
n
n
n
z
c x
=
=
.
2
1
1
2
3
2
1
2
3
2
2
2
2
3
4
2
'
2
3
... (
2)
...
'
2
3
... (
2)
...
''
2
2 3
3 4
... (
1)(
2)
...
n
x
n
x
n
x
z
c
c x
c x
n
c
x
z
c
c
c x
n
c
x
x
x
z
c
c x
c x
n
n
c
x
+
+
+
+
= +
+
+ +
+
+
=
+
+
+ +
+
+
=
+
+
+ +
+
+
+
Substituting the resulting series into the equation, we obtain the following equality:
1
2
2
0
3
3
1
4
2
4
2
2
2
2
1
[2
(2
1)2
] [2 3
(2
1)3
]
[3 4
(2
1)4
]
... [(
1)(
2)
(2
1)(
2)
]
...
0
n
n
n
n
v
c
c
v
c
c
c
v
c
c x
c
x
v
c
c
x
n
n
c
v
n
c
c x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ +
+
+
+
+
+
+
+ =
According to the method of undetermined coefficients, we set all coefficients of powers of
x
equal to zero. This
is because for the series sum to be zero, each coefficient must be zero.
0
x
for
1
0
c
=
(3)
2
2
(
1)(
2)
(2
1)(
2)
0
n
n
n
n
n
c
v
n
c
c
+
+
+
+
+
+
+
+ =
1,2,...
n
=
From this, we obtain the following recurrence formula:
2
(
2)(
2
2)
n
n
c
c
n
n
v
+
= −
+
+
+
0, 1, 2,...
n
=
(4)
Based on
(3)
and
(4)
:
American Journal of Applied Science and Technology
28
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
1
3
5
2
1
...
... 0
n
c
c
c
c
−
= = = =
= =
0
2
2(2
2)
c
c
v
= −
+
2
0
4
4(2
4)
2 4(2
2)(2
4)
c
c
c
v
v
v
= −
=
+
+
+
......................................................
0
2
0
2
( 1)
2 4 ... 2
(2
2)(2
4) ... (2
2 )
( 1)
2
1 2 ...
(
1)(
2) ... (
)
n
n
n
n
c
c
n
v
v
v
n
c
n
v
v
v
n
= −
=
+
+
+
= −
+
+
+
Thus, the solution of the equation is represented by the series:
2
0
2
1
( 1)
1
2
1 2 ...
(
1)(
2) ... (
)
n
n
n
n
x
z
c
n v
v
v
n
=
−
=
+
+
+ +
(5)
Usually, the constant
0
c
is chosen as:
0
1
2
(
1)
v
c
Г n
=
+
The Gamma function is considered a generalization of the factorial:
1 2 ...
!
(
1)
n
n
Г n
= =
+
(
1)(
2) ... (
) (
1)
(
2)(
3) ... (
) (
2)
(
) (
)
(
1)
v
v
v
n Г v
v
v
v n Г v
v
n Г v
n
Г v n
+
+ +
+ =
+
+ +
+
=
=
+
+
=
+ +
We can write the function in the following form:
2
2
1
2
2
0
1
( 1)
2
(
1)
2
1 2 ...
(
1)(
2) ... (
)
(
1)
( 1)
2
(
1)
(
1)
n
n
v
n v
n
n
n
n v
n
x
z
Г n
n v
v
v
n Г v
x
Г n
Г v
n
+
=
+
=
−
=
+
=
+
+
+
+
+
−
=
+
+ +
The solution of the equation consists of the function. We denote this function as
( )
v
J x
. Therefore,
2
0
( 1) (
2)
( )
(
1) (
1)
v
n
n v
n
x
J
x
Г n
Г v n
+
=
−
=
+
+ +
This function is called the Bessel function of the first kind
of index
v
or order
v
In some literature, these functions
are also referred to as cylindrical functions.
The Bessel functions exhibit sinusoidal-like behavior.
( )
v
J x
has periodic oscillations similar to sine and
cosine, but these oscillations gradually increase and
decrease in amplitude.
American Journal of Applied Science and Technology
29
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
The above graph shows Bessel functions of the first
kind
( )
n
J
x
for different
0, 1, 2, 3, 4, 5
n
=
values. The important characteristics of this graph are
reflected in mathematical, physical, and engineering
applications. As evident from the graph, the amplitude
(maximum value) of each
( )
n
J
x
function decreases
as
x
−
increases. This phenomenon reflects the
relationship of Bessel functions to resonance systems
in physics and engineering.
CONCLUSION
Bessel functions play a crucial role in mathematical
analysis and physics. Their derivation using power
series method and fundamental properties make them
widely applicable in solving numerous theoretical and
practical problems. The graphical analysis of Bessel
functions and their relationships with other special
functions enable the development of new research and
practical applications. Bessel functions maintain their
significance in mathematical analysis and physics.
REFERENCES
Salohiddinov, M.S. (2007). Integral Equations.
Tashkent. 256 p.
Kreh, M. (n.d.). Bessel Functions. Pennsylvania State
University.
Retrieved
from:
http://www.math.psu.edu/papikian/Kreh.pdf
Sansone, G. (1959). Orthogonal Functions. New York,
NY: Interscience Publishers Inc.
Fillipov, A.M. (1985). Collection of Problems in
Differential Equations. Moscow: Nauka. 128 p.
Mamatov, M. (1995). Collection of Problems from
Differential Equations. Tashkent: University. 156 p.
