МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
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Том 2, Выпуск 12,
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“COMPLEX NUMBERS AND THEIR CONNECTION WITH ANALYTIC
FUNCTIONS”
Khudaikulova Saida Zakirovna
Teacher of Termez State Pedagogical Institute
Phone: +99890-246-47-47
Anvarova Nigina Fayzulloevna
2nd-year student of
Temez State Pedagogical Institute
Annotation
: This paper explores the relationship between complex numbers
and analytical functions. Complex numbers consist of real and imaginary parts
and are widely used in various fields of mathematics. Analytic functions are
crucial when working with complex variables. These functions are based on the
Cauchy-Riemann equations, which define their properties in a given domain. The
study examines the key concepts of complex functions, their differentiability, and
analytical properties.
Keyword
: Complex numbers, Analytic function, Cauchy-Riemann equations,
Holomorphic functions, Contour integral, Differentiability, Taylor series,
Analysis, Complex analysis, Existence of analytic function
Complex numbers and their analytic functions are one of the main areas of
mathematics and are widely used in many fields such as physics, engineering,
and others. Let me provide a detailed explanation about complex numbers and
their analytic functions. Complex numbers are actually a combination of two real
numbers. Every complex number is written in the following form:
z=x+yi
Here:
•
x
is the real part,
•
y
is the imaginary part,
•
i
is the imaginary unit, defined as
i² = −1
.
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
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Том 2, Выпуск 12,
31
Декабрь
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Complex numbers have a real part and an imaginary part, often denoted as xx
and yy, respectively. Complex functions are functions that accept real variables
and convert them into complex numbers. The general form of a complex function
can be written as:
f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y)
In the theory of complex numbers, an analytic function (or holomorphic
function) is a function that is differentiable with respect to a complex variable,
and it satisfies the following conditions:
1.
The function must be continuous and differentiable.
2.
The Cauchy-Riemann equations must be fully satisfied.
For a complex function to be analytic, the Cauchy-Riemann equations must
hold, and they are expressed as:
If f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y), where u(x,y)u(x, y) and
v(x,y)v(x, y) are the real and imaginary parts, respectively, then the Cauchy-
Riemann equations are:
These equations link the changes in
u
and
v
, and only when these conditions
are satisfied does the function become analytic. If a complex function is analytic,
its derivative also exists and is continuous. This is different from real functions,
where the derivative may only exist at certain points, but for complex functions,
analyticity means the derivative exists at every point.
Complex numbers and analytic functions are widely used in various fields
such as electromagnetic waves, quantum mechanics, mathematical physics, and
signal processing. They play a crucial role in modeling system dynamics and
solving linear differential equations. Complex functions also have significant
importance in tools like integral transforms, Laplace transforms, Fourier
transforms, and others. These tools can be applied more efficiently using analytic
functions.
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
291
https://universalpublishings.com
In general, complex numbers and their analytic functions are of great
importance not only in mathematics but also in the application of science and
technology.
References:
1.
Xudaykulova, S. (2024). DARAJALI GEOMETRIYA - KO‘PHADLAR VA
NORMAL KONUSLAR. Interpretation and Researches, 1(1). извлечено от
https://interpretationandresearches.uz/index.php/iar/article/view/2496
Mathematical Analysis
(Yu. M. Geller, L. D. Faddeev).
2.
Xudaykulova , S. (2024). TEXNIK IJODKORLIKNING HOZIRGI HOLATI.
Research
and
Implementation.
извлечено
от
journal.uz/index.php/rai/article/view/520
3.
Ne’matova , D. (2023). BOSHLANG‘ICH SINF O‘QUVCHILARIDA
TANQIDIY FIKRLASH KO‘NIKMALARINI SHAKLLANTIRISHNING
PEDAGOGIK-PSIXOLOGIK
XUSUSIYATLARI.
Interpretation
and
Researches,
2(1).
извлечено
от
https://interpretationandresearches.uz/index.php/iar/article/view/973
4.
Холмуминова, А. (2023). ОСОБЕННОСТИ И ПРЕИМУЩЕСТВА
ФОРМИРОВАНИЯ
КОМПЕТЕНТНОСТИ
ПОДГОТОВКИ
ИННОВАЦИОННОЙ ДЕЯТЕЛЬНОСТИ У БУДУЩИХ УЧИТЕЛЕЙ
НАЧАЛЬНЫХ КЛАССОВ. Interpretation and Researches, 2(1). извлечено
от
https://interpretationandresearches.uz/index.php/iar/article/view/1145
"Comple
x
Analysis"
by
Elias
M.
Stein
and
Rami
Shakarchi
This book introduces the key concepts, theorems, and practical applications of
complex analysis.
5.
"Principles
of
Mathematical
Analysis"
by
Walter
Rudin
Rudin's famous "Mathematical Analysis" book contains important material on
complex numbers.
6.
"Complex Variables and Applications" by James Brown and Ruel Churchill
A classic textbook that explains complex variables and their practical
applications in detail.
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
292
https://universalpublishings.com
7.
"Introduction to Complex Analysis" by Richard A. Silverman
