THE DOMAINS OF CONVERGENCE OF THE TAYLOR SERIES

Аннотация

В этой статье представлены некоторые концепции разложения функций в ряд Тейлора при некотором значении. Кроме того, представлены сходимости рядами Тейлора и анализ области сходимости. Здесь мы опишем некоторые из простейших примеров областей в пространстве 𝐶𝑛. Как обычно, область - это открытое связное множество, где открытость означает, что вместе с любой его точкой множество также содержит окрестность этой точки, а связность открытого множества 𝐷 означает, что для любых точек 𝑧′,𝑧» ∈ 𝐶𝑛 существует непрерывное множество 𝛾:[0,1] → 𝑑 для которого 𝛾(0) = 𝑧ʹ и 𝛾(1) = 𝑧ʹʹ. Ключевые слова. Степенный ряды, комплексные числа, несколько переменных, точка, функция. THE DOMAINS OF CONVERGENCE OF THE TAYLOR SERIES Kh.G.Khayitova – teacher of the «Mathematical Analysis» department, Bukhara State University, xilola_xayitova@mail.ru, x.g.xayitova@buxdu.uz. https://scientific-jl.com/luch/ 2 1 0 Часть-38_ Том-2_ Февраль-2025 Annotation. This article presents some concepts of Taylor series expansion of functions for some value. In addition, Taylor series convergence and convergence domain analysis are presented. Here we shall describe some of the simplest examples of domains in the space 𝐶𝑛. As usual, a domain is an open connected set, where openness means that along with any point of it the set also contains a neighborhood of that point, and connectedness of an open set 𝐷 means that, for any points 𝑧ʹ,𝑧ʹʹ𝜖 𝐷 there exists a continuous are 𝛾:[0,1] → 𝑑 for which 𝛾(0) = 𝑧ʹ and 𝛾(1) = 𝑧ʹʹ. Key words: degree series, several variables, complex numbers, point, function. One of the main theorems of the theory of a complex variable is Theorem: Let 𝑓 𝜖 𝑂(𝐷) and let 𝑧0 𝜖 𝐷 be an arbitrary point in 𝐷. Then the function 𝑓 may be represented as a sum of a convergent power series 𝑓(𝑧) = ∑𝑐 (𝑧 − 𝑧 )𝑛 𝑛 0 𝑛=0 inside any disk 𝑈 = {ǀ𝑧 − 𝑧 ǀ 0 so that |𝑧 − 𝑧_0 | < 𝑟 < 𝑅 and denote by {𝛾 = ϛ ∶ |ϛ − 𝑧| = 𝑟}. The integral Cauchy formula implies that 𝑟 1 𝑓(ϛ) 𝑓(𝑧) = ∫ 𝑑ϛ. 2𝜋𝑖 ϛ − 𝑧 𝐷 in order to represent f as a power series let us represent the kernel of this integral as the sum of a geometric series: 1 𝑧 − 𝑧 −1 (𝑧 − 𝑧 )𝑛 0 0 = [(ϛ − 𝑧 )(1 − )] = ∑ . ϛ − 𝑧 0 ϛ − 𝑧 (ϛ− 𝑧 )𝑛+1 0 0 𝑛=0 https://scientific-jl.com/luch/ 2 1 1 Часть-38_ Том-2_ Февраль-2025 1 We multiply both sides by 𝑓(ϛ) and integrate the series term-wise along 𝛾 . 𝑟 2𝜋𝑖 The series converges uniformly on 𝛾 since 𝑟 𝑧 − 𝑧 |𝑧 − 𝑧 | 0 0 | | = = 𝑞 < 1 ϛ − 𝑧 𝑟 0 for all ϛ𝜖𝛾 . Uniform convergence is preserved under multiplication by a 𝑟 1 continuous and hence bounded function 𝑓(ϛ). Therefore our term-wise 2𝜋𝑖 integration is legitimate and we obtain [1-7] 1 𝑓(ϛ)𝑑ϛ 𝑓(𝑧) = ∫ (𝑧 − 𝑧 )𝑛 = ∑𝑐 (𝑧 − 𝑧 )𝑛 2𝜋𝑖 (ϛ− 𝑧 )𝑛+1 0 𝑛 0 0 𝐷 𝑛=0 where 1 𝑓(ϛ)𝑑ϛ 𝑐 = ∫ 𝑛 = 0,1,… 𝑛 2𝜋𝑖 (ϛ− 𝑧 )𝑛+1 0 𝐷 Definition: The power series with coefficient given by is the Taylor series of the function 𝑓 at the point 𝑧 (or centered at 𝑧 ). 0 0 The Cauchy theorem implies that the coefficients 𝑐 of the Taylor series 𝑛 defined by do not depend on the radius 𝑟 of the circle 𝛾 ,0 < 𝑟 < R. 𝑟 Exercise 1. Find the radius of the largest disk where the function 𝑧 ⁄ 𝑠𝑖𝑛 𝑧 may be represented by a Taylor series centered at 𝑧 = 0. 0 Exercice 2. Let f be holomorphic in 𝐶. Show that (𝑎) 𝑓 is even if and only if its Taylor series at 𝑧 = 0 contains only even powers; (𝑏) 𝑓 is real on the real axis if and only if 𝑓(𝑧̅) = 𝑓(𝑧) for all 𝑧 ∈ 𝐶. We present some simple corollaries of Theorem [7-13]. The Cauchy inequalities. Let the function 𝑓 be holomorphic on a closed disk U̅ = {|𝑧 − 𝑧 | ≤ 𝑟} and let its absolute value on the circle 𝛾 = 𝜕U be 0 𝑟 bounded by a constant 𝑀. then the coefficients of the Taylor series of 𝑓 at 𝑧 satisfy 0 the inequalities |𝑐| ≤ M⁄𝑟𝑛 (𝑛 = 0,1,…) https://scientific-jl.com/luch/ 2 1 2 Часть-38_ Том-2_ Февраль-2025 Proof. We deduce from exercise 1 using the fact that |𝑓(ϛ)| ≤ M for all ϛ𝜖𝛾 : 𝑟 1 M M |𝑐 | ≤ 2𝜋𝑟 = . 𝑛 2𝜋𝑟𝑛+1 𝑟𝑛 Exercise 3. Let 𝑃(𝑧) be a polynomial in 𝑧 of degree 𝑛. Show that if |𝑝(𝑧)| ≤ M for |𝑧| = 1 then |𝑝(𝑧)| ≤ m|𝑧|n for all |𝑧| ≥ 1. The Cauchy inequalities simply the interesting. Theorem . (Liouville) If the function 𝑓 is holomorphic in the whole complex plane and bounded then it is equal identically to a constant. Proof. According to theorem the function 𝑓 may be represented by a Taylor series 𝑓(𝑧) = ∑ 𝑐 𝑧𝑛 𝑛 𝑛=0 in any closed disk U̅ = {|𝑧| ≤ 𝑟}, 𝑟 < ∞ with the coefficients that do not depend on 𝑅. Since f is bounded in 𝐶, say |𝑓(𝑧)| ≤ 𝑀 then the Cauchy inequalities imply that for any 𝑛 = 0,1,…. we have |𝑐 | ≤ M⁄R𝑛. We may take 𝑅 to be arbitrary 𝑛 large and hence the right side tends to zero as 𝑅 → +∞ while the left side is independent of 𝑅. Therefore the two properties of a function-to be holomorphic and bounded are realized simultaneously only for the trivial functions that are equal identically to a constant [10-25]. Theorem: If a function 𝑓 is holomorphic in the closed complex plane 𝐶 then it is equal identically to a constant. Proof. If the fun