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109
Namangan davlat pedagogika institute Intellektual fanlar va axborot texnologiyalari
oktamaliyevikromjon@gmail.com
Namangan davlat pedagogika instituti Matematika-informati
-bosqich
talabasi
Annotatsiya: "Fur'e qatori" matematik tahlil va funktsional analiz sohasida keng
qo'llaniladigan, maxsus bir qatorlar bo'lib, uzluksiz yoki diskret funksiyalarni
o'zgaruvchilarga qarab ifodalash imkonini beradi. Ularning asosiy xususiyati shundaki,
ular yordamida murakkab tizimlar yoki funksiyalarni osonroq tahlil qilish mumkin.
Qatorning umumiy shakli odatda trigonometrik funksiyalar (sinus, kosinus) yoki boshqa
analitik funksiyalar asosida quriladi. Fur'e qatorlari fizik hodisalarni modellashda,
signalni tahlil qilishda va boshqa ko'plab ilmiy sohalarda muhim ahamiyatga ega.
qator, yaqinlashish, tekis yaqinlashish.
METHODS FOR FINDING THE SUMS OF CERTAIN NUMERICAL SERIES
USING FOURIER SERIES
Teacher at the Department of Intellectual Sciences and Information Technologies,
Namangan State Pedagogical Institute
oktamaliyevikromjon@gmail.com
Jurakhon
Second-year student at Namangan State Pedagogical Institute
Annotation: The Fourier series is an important tool in mathematical analysis that allows
functions or systems to be expressed in terms of trigonometric functions. With the help of
this series, complex functions can be represented as the sum of simple trigonometric
functions (sine, cosine). Fourier series are widely used in physics, signal analysis, the
study of sound and light waves, and many other scientific and engineering fields. They
provide a more convenient and efficient way to analyze systems and processes.
Keywords: Fourier series, Fourier series of even and odd functions, trigonometry,
numerical series, approximation, smooth approximation.
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110
oktamaliyevikromjon@gmail.com
A
KIRISH
1 .Definition of Fourier series. Each term
consists of harmonics, and the following functional series:
is called a trigonometric series. Here, the coefficients
are referred to as the coefficients of the trigonometric series. Typically, the partial sum
of the trigonometric series (1) is expressed as:
and is referred to as the trigonometric polynomial.
Let
be a function defined on the interval
In the series (2), we introduce the
following notations:
0
1
1
1
( )
,
( ) cos(
)
,
( )sin(
)
, (
)
k
k
a
f x dx a
f x
kx dx b
f x
kx dx
k
N
.
Substituting the notations (3) into the trigonometric series (1), as
the
trigonometric series (2) becomes the
Fourier series
of the function
on the interval
. It is known that in this case, if
is a continuous function on
, then
its Fourier series converges to the function itself. [1. B.200]
Thus, the following equality can be observed in this case:
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2. Fourier Series for Even and Odd Functions.
Let
be an even function defined
on
, and assume it is integrable on this interval. We will determine the Fourier
coefficients of this function:
Thus, the Fourier coefficients of the even function
are:
and the Fourier series is:
Now, suppose
is an odd function defined on
, and assume it is integrable on
this interval. The Fourier coefficients for this function are as follows:
and the Fourier series is:
3. Applications of Fourier Series to Numerical Series Calculations
Example 1: Compute the sum of the series
1
1
( 1)
2
1
n
n
n
.
Solution: First, we find the Fourier series of the function
defined on
To
do this, we calculate the Fourier coefficients of the given function:
.
This simplifies to:
1
2
1
sin
n
n
f x
x
nx
n
Since
is continuous on the interval
, we can express the Fourier series as:
Substituting
into this equality, we get:
:
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112
The equality becomes:
Answer:
Example 2: Compute the sum of the series
Solution: First, we find the Fourier series of the function
defined on
To do this, we calculate the Fourier coefficients of the given function.
2
2
0
0
2
2
3
a
x dx
2
2
0
0
2
0
0
0
2
2
sin
4
4
cos
1
4
cos
sin
cos
1
n
n
nx
x
nx
a
x
nxdx
x
x
nxdx
nxdx
n
n
n
n
n
n
The Fourier series for
is given as:
2
2
2
1
cos
4
1
3
n
n
nx
f x
x
n
Since
is continuous on the interval
we can write:
.
Substituting
into the series, we get:
Answer:
Example 3: Compute the sum of the series
Solution: First, we determine the Fourier series of the function
defined on
To do so, we calculate the Fourier coefficients.
For the coefficient
, we compute:
From Example 2, we know:
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Thus, the Fourier series of
becomes:
Since
is continuous on
we can write:
Setting
, we obtain:
Substituting back, we find:
From Example 1, we know:
Answer:
Example 4: Compute the sum of the series
Solution: First, we determine the Fourier series of the function
defined on
. To do this, we compute the Fourier coefficients.
4
4
0
0
2
2
5
a
x dx
For
, we calculate:
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From Example 3, we know:
Thus, we have
The Fourier series of
then becomes:
Since
is continuous on
, we write:
Setting
, we obtain:
From Example 2, we know:
Substituting this result:
Answer:
.
CONCLUSION AND RECOMMENDATIONS
In this article, we explored the application of Fourier series in evaluating the sums of
certain infinite numerical series. By expressing periodic functions as Fourier series, we
demonstrated how analytical techniques can simplify the computation of otherwise
complex series. The method not only provides elegant derivations of well-known results
but also reveals deeper insights into the structure of numerical series through harmonic
analysis. These examples highlight the power and versatility of Fourier series in
mathematical problem-solving, especially in the field of real analysis.
Recommendations:
1.
Further study of Fourier series applications in solving problems in physics
and engineering is recommended, as these fields often involve periodic phenomena.
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2.
It is beneficial to explore the convergence criteria of Fourier series in more
detail, especially in cases where the function has discontinuities or is not piecewise
smooth.
3.
Future work can include numerical simulations or visualizations of Fourier
approximations to better understand the behavior of series convergence.
4.
Investigating other orthogonal expansions, such as Fourier Bessel or
Legendre series, may provide additional tools for analyzing different classes of functions
and series.
References
1.
G.
Xudoyberganov, A. K. Vorisov, X. T. Mansurov, B.A.Shoimqulov, Matematik
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A. G`oziyev, I. Isroilov, M. Yaxshiboyev, Matematik analizdan misol va masalalar I,
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