## 数学代写|复分析作业代写Complex function代考|MATH2242

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Related Studies and Applications

The AFD type expansions is in a great extent related to the Beurling-Lax shiftinvariant subspaces of the Hardy $H^2$ spaces. In the unit disc case,
$$H^2(\mathbf{D})=\overline{\operatorname{span}}\left{B_k\right}_{k=1}^{\infty} \oplus \phi H^2(\mathbf{D}),$$
where $\left{B_k\right}_{k=1}^{\infty}$ is the TM system generated by a sequence $\left{a_1, \cdots, a_n, \cdots\right}$, where multiples are counted, and $\phi$ is the Blaschke product with the zeros $\left{a_1, \cdots, a_n, \cdots\right}$ including the multiples. Note that when a Blaschke product $\phi$ having $a_k$ ‘s as all its zeros does not exist, corresponding to the condition
$$\sum_{k=1}^{\mathbf{1 2})}\left(1-\left|a_k\right|\right)<\infty,$$
then the associated TM system is a basis. Although this has been well known over a long time, its relations with adaptive expansions, as far as what are aware by the author, have not been brought up. The fact that TM systems being Schauder systems was proved in [93]. The space decomposition relation (26) was extended to $H^p$ spaces, where $p \neq 2$ [80]. Relations between backward shift invariant subspaces and bandlimited functions and Bedrosian identity $[80,107]$ were studied. There are open questions on whether there exist adaptive and fast converging expansions by using TM systems for the cases $p \neq 2$, and for $p=2$ how far one can extend AFD (26) to higher dimensions. The study has a great room to be further developed.

## 数学代写|复分析作业代写Complex function代考|Extra-Strong Uncertainty Principle

The phase and frequency studies in mono-component function theory lay certain foundations in digital signal processing. In related studies what is called extra-strong uncertainty principle
$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left(\int _ { – \infty } ^ { \infty } \left|t-\langle t\rangle\left|\left.|\phi(t)-\langle\omega\rangle || f(t)\right|^2 d t\right)^2\right.\right.$$
was recently established [22], where $f$ is a real-valued signal, $\sigma_t^2$ and $\sigma_\omega^2$ are the standard deviations with respect to the time and the Fourier frequency, and $\langle t\rangle$ and $\langle\omega\rangle$ are the corresponding means. A weaker uncertainty principle of the same type was previously given by L. Cohen
$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left.\left.\left|\int_{-\infty}^{\infty}(t-\langle t\rangle)(\phi(t)-\langle\omega\rangle)\right| f(t)\right|^2 d t\right|^2$$
[13]. We further extended the above result to multi-dimensional contexts [21-24, 26].

The Dirac-type time-frequency distribution (DTFD) of the form
$$P(t, \omega)=\rho^2(t) \delta\left(\omega-\theta^{\prime}(t)\right)$$
is the ultimate desire of signal analysts. Several time-frequency distributions, including windowed Fourier transform and Wigner-Ville transform, etc., have been used by signal analysts, of which none are entirely satisfied. The existing timefrequency distributions do not give explicit and clear frequency components, and, they often depend on parameter selections. Positive-frequency decompositions of signals offered by the AFD decompositions naturally give rise to Dirac-type timefrequency distributions. For a single mono-component $m_1(t)=\rho_1(t) \cos \theta_1(t)$ the corresponding DTFD according to (28) is the graph $\left(t, \theta_1^{\prime}(t)\right)$ of the function $\omega=\theta_1^{\prime}(t)$ in the $\omega-t$ plane, while the weight $\rho_1^2(t)$ may be represented by colors continuously changing along with changing of the values $\rho_1^2(t)$. If a signal $f$ is expanded into a series of “intrinsic composing” mono-components, then its DTFD is the bunch of color-weighted graphs of which each is made from a composing monocomponent $[20,126]$. This definition has been interested and being paid attention by signal analysts including Leon Cohen and Lorenzo Galleani, etc., and has been used in practice (see below the application section).

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Related Studies and Applications

AFD 类型展开在很大程度上与 Hardy 的 Beurling-Lax 位移不变子空间相关 $H^2$ 空间。在单元盘盒中，
$\mathrm{H}^{\wedge} 2(\backslash \mathrm{mathbf}{\mathrm{D}})=$ loverline ${\backslash 0$ peratorname ${$ span $}} \backslash l$ eft $\left{B_{-} k \backslash \text { ight }\right}_{-}{\mathrm{k}=1}^{\wedge}{\backslash$ infty $} \backslash$ plus $\backslash p h i \mathrm{H}^{\wedge} 2(\backslash \mathrm{mathbf}{\mathrm{D}})$, , 算倍数，并且 $\phi$ 是零点的 Blaschke 积 Ileft{a_1, Icdots，a_n, Icdots\right } } \text { 包括倍数。请注意，当 Blaschke } 产品 $\phi$ 有 $a_k$ 的因为它的所有雩都不存在，对应于条件
$$\sum_{k=1}^{12)}\left(1-\left|a_k\right|\right)<\infty$$

## 数学代写|复分析作业代写Complex function代考|Extra-Strong Uncertainty Principle

$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left(\int_{-\infty}^{\infty}|t-\langle t\rangle||\phi(t)-\langle\omega\rangle||f(t)|^2 d t\right)^2$$

$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left.\left.\left|\int_{-\infty}^{\infty}(t-\langle t\rangle)(\phi(t)-\langle\omega\rangle)\right| f(t)\right|^2 d t\right|^2$$
[13]. 我们进一步将上述结果扩展到多维上下文 [21-24, 26]。

$$P(t, \omega)=\rho^2(t) \delta\left(\omega-\theta^{\prime}(t)\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|MATH307

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Cyclic AFD for n-Best Rational Approximation

In core-AFD the parameters $a_1, \ldots, a_n, \ldots$ are selected in the one by one manner to obtain an optimal sequence of Blaschke forms to approximate the given function
$$\sum_{k=1}^n\left\langle f, B_{\left{a_1, \cdots, a_k\right}}\right\rangle B_{\left{a_1, \cdots, a_k\right}}(z) .$$
Now we change the question to the following: Given $f \in H^2(\mathbf{D})$ and a fixed positive integer $n$, find $n$ parameters $\tilde{a}1, \ldots, \tilde{a}_n$ such that the associated $n$-Blaschke form best approximates $f$, that is, \begin{aligned} & \left|f-\sum{k=1}^n\left\langle f, B_{\left{\tilde{a}1, \cdots, \tilde{a}_k\right}}\right\rangle B{\left{\tilde{a}1, \cdots, \tilde{a}_k\right}}(z)\right| \ = & \min \left{\left|f-\sum{k=1}^n\left\langle f, B_{\left{b_1, \cdots, b_k\right}}\right\rangle B_{\left{b_1, \cdots, b_k\right}}(z)\right|:\left{b_1, \cdots, b_n\right} \in \mathbf{D}^n\right} . \end{aligned}
This amounts an optimization with simultaneous selected $n$ parameters that is obviously better than one on selections of $n$ parameters in the one by one manner. Simultaneous selection of the parameters in an approximating $n$-Blaschke form is equivalent with the so-called optimal approximation by rational functions of degrees not larger than $n$. The latter problem was phrased as $n$-best rational approximation. It has been a long standing open problem, presented as follows.

Let $p$ and $q$ denote polynomials of one complex variable. We say that $(p, q)$ is an n-pair if $p$ and $q$ are co-prime, both of degrees less than or equal to $n$, and $q$ does not have zero in the unit disc. Denote by $\mathcal{R}_n$ the set of all such $n$-pairs. If $(p, q) \in \mathcal{R}_n$, then the rational function $p / q$ is said to be a rational function of degree less or equal $n$. Let $f$ be a function in the Hardy $H^2$ space in the unit disc. To find an $n$-best rational approximation to $f$ is to find an $n$-pair $\left(p_1, q_1\right)$ such that
$$\left|f-p_1 / q_1\right|=\min \left{|f-p / q|:(p, q) \in \mathcal{R}_n\right} .$$
Existence of such a minimum solution was proved many decades ago [4, 112], a practical algorithm to get a solution, however, has been an open problem till now. The best $n$-Blaschke form approximation is essentially equivalent with the $n$-best rational approximation. There are separate proofs for existence of the solution of optimization problem (15) $[75,84]$. By taking advantages of the explicit form and the orthogonality of Blaschke forms we get a practical algorithm for the classical $n$-best rational approximation problem.

## 数学代写|复分析作业代写Complex function代考|Pre-Orthogonal Adaptive Fourier Decomposition

The approximation theory and algorithm that were developed in the previous sections can be extended to more general contexts. To explain just the idea we restrict ourselves to the simplest cases, including the weighted Bergman spaces and weighted Hardy spaces, etc. Assume that Hilbert space $\mathcal{H}$ consists of functions defined in an open connected region $\mathcal{E}$ (can be unbounded) in the complex plane, and the reproducing kernel $k_a$ is an analytic function of the variable $a$ in $\mathcal{E}$ satisfying the relation
$$f^{(l)}(a)=\left\langle f,\left(\frac{\partial}{\partial \bar{a}}\right)^l k_a\right\rangle, \quad l=1,2, \cdots$$
Let $\left{a_1, \cdots, a_n, \cdots\right}$ be a finite or infinite sequence. For a fixed $n$ we define the multiple of $a_n$, denoted by $l\left(a_n\right)$, to be the repeating times of $a_n$ in the $n$-tuple $\left{a_1, \cdots, a_n\right}$. With this definition, for instance, the multiple of $a_1$ is just 1 , and the multiple of $a_2$ will depend on whether $a_2=a_1$. If yes, then $l\left(a_2\right)=2$, and, if not, $l\left(a_2\right)=1$, and so on. Note that it is a little abuse of notation for it is not dependent on the value of $a_n$ but on the repeating times of $a_n$ in the corresponding $n$-tuple. We accordingly define
$$\tilde{k}{a_n} \triangleq\left[\left(\frac{\partial}{\partial \bar{a}}\right)^{l\left(a_n\right)-1} k_a\right]{a=a_n} \triangleq\left(\frac{\partial}{\partial \bar{a}}\right)^{l\left(a_n\right)-1} k_{a_n} .$$
We further assume the following boundary vanishing condition, implying the maximal selection principle in every individual context, as follows: Let $a_1, \cdots, a_{n-1}$ be previously given, and $\left{B_1, \cdots, B_{n-1}\right}$ be the Gram-Schmidt orthonormalization of $\left{\tilde{k}{a_1}, \cdots, \tilde{k}{a_{n-1}}\right}$, then for every $f \in \mathcal{H}$, the pre-orthogonal system has the property
$$\lim _{a \rightarrow \partial \mathcal{E}}\left\langle f, B_n^a\right\rangle=0,$$ where $\left{B_1, \cdots, B_{n-1}, B_n^a\right}$ is the Gram-Schmidt orthonormalization of $\left{\tilde{k}{a_1}, \cdots, \tilde{k}{a_{n-1}}, k_a\right}$, with $a \neq a_k, k=1, \cdots, n-1$. We note (1) if $a \rightarrow \partial \mathcal{E}$, then $a$ is different from any already selected $a_k, k=1, \cdots, n-1$; and (2) in any case the limit $a \rightarrow \partial \mathcal{E}$ is in the sense of the topology of the one-point-compactification of the complex plane while the “one point” takes to be $\infty$. With boundary vanishing assumption we conclude the maximal selection principle of POAFD: Under the assumption (21), through a compact argument, there exists a sequence $\left{b_j\right}_{j=1}^{\infty}$ such that none of the $b_j$ ‘s take any values $a_1, \cdots, a_{n-1}$, and $\lim {j \rightarrow \infty} b_j \triangleq a_n \in \mathcal{E}$, and $$\lim {j \rightarrow \infty}\left|\left\langle f, B_n^{b_j}\right\rangle\right|=\max \left{\left|\left\langle f, B_n^a\right\rangle\right|: a \in \mathcal{E}\right}$$

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Pre-Orthogonal Adaptive Fourier Decomposition

$$f^{(l)}(a)=\left\langle f,\left(\frac{\partial}{\partial \bar{a}}\right)^l k_a\right\rangle, \quad l=1,2, \cdots$$

$$\tilde{k} a_n \triangleq\left[\left(\frac{\partial}{\partial \bar{a}}\right)^{l\left(a_n\right)-1} k_a\right] a=a_n \triangleq\left(\frac{\partial}{\partial \bar{a}}\right)^{l\left(a_n\right)-1} k_{a_n} .$$

$$\lim {a \rightarrow \partial \mathcal{E}}\left\langle f, B_n^a\right\rangle=0,$$ 在哪里 Veft{B_1, \cdots, B{n-1}, B_n^a\right } } \text { 是 Gram-Schmidt 正交化 }
Veft{\tilde{k}{a_1}, \cdots, \tilde{k}{a_{n-1}}, k_a\right} ， 和 $a \neq a_k, k=1, \cdots, n-1$. 我们注意到 (1) 如 果 $a \rightarrow \partial \mathcal{E} ，$ 然后 $a$ 不同于任何已选择的 $a_k, k=1, \cdots, n-1$ ； (2) 在任何情况下限制 $a \rightarrow \partial \mathcal{E}$ 是在复 平面的一点紧化的拓扑意义上，而“一点”是 $\infty$. 通过边界消失假设，我们得出 POAFD 的最大选择原则： 在假设 (21) 下，通过坚凑的论证，存在一个序列 \eft{b_jright $\left.}_{-}{j=1}^{\wedge}{\backslash i n f t y}\right}$ 竝样就没有 $b_j$ 取任何值 $a_1, \cdots, a_{n-1}$ ， 和lim $j \rightarrow \infty b_j \triangleq a_n \in \mathcal{E} ，$ 和

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|KMA152

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|One Dimensional Core-Adaptive Fourier Decomposition

Due to the above-mentioned reason we decide to use the rational orthonormal system, or by another name the Takenaka-Malmquist, or TM system in brief, introduced in Theorem 2.2. We note that TM systems in general cannot be avoided for they are Gram-Schmidt (G-S) orthogonalization of the partial fractions with poles outside the closed unit disc, the latter being fundamental constructive building blocks of rational functions in the Hardy spaces. TM systems consist of functions of positive frequency due to their construction in (finite) Blaschke products. The difference between our use and the traditional use of TM systems is that we make the parameters defining the system to be adaptive: For every individual function or signal we expand it by using a suitable TM system while the determining parameters are deliberately selected according to the data of the given function. The TM system itself may not be a basis. Whether or not the system in use is a basis, is, in fact, not interested or required. On the other hand, the adaptive expansion in the selected TM system converges very fast. And, additionally, each expanding term has positive non-constant and non-linear instantaneous frequencies. In contrast, the traditional use of a TM system is based on a fixed collection of parameters making the corresponding TM system a basis of the underlying space. The reason of use of a particular and fixed collection of parameters, however, is, as usual, not be well justified. Laguerre and two-parameter Kautz systems are examples of such fixedparameter TM bases.

In the sequel we change our function notation $s^{+}$in the Hardy $H^2(\mathbf{D})$ to $f$. In the unit circle context we have $f(z)=\sum_{l=1}^{\infty} c_l z^l, \sum_{l=1}^{\infty}\left|c_l\right|^2<\infty$. Now we seek a decomposition of $f$ into a TM system with adaptively selected parameters. The collection of the functions
$$e_a(z)=\frac{\sqrt{1-|a|^2}}{1-\bar{a} z}, \quad a \in \mathbf{D},$$
consists of normalized Szegö kernels of the disc. Below we present AFD, or more specifically, Core-AFD algorithm. Set $f=f_1$. First write
$$f(z)=\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)+\frac{f_1(z)-\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)}{\frac{z-a_1}{1-\bar{a}_1 z}} \frac{z-a_1}{1-\bar{a}_1 z}$$

## 数学代写|复分析作业代写Complex function代考|Unwinding AFD

Let $f=h g$, where $f, g$ are Hardy $H^2(\mathbf{D})$ functions, and $h$ is an inner function. Let $f$ and $g$ be expanded into their respective Fourier series, viz.,
$$f(z)=\sum_{k=0}^{\infty} c_k z^k, \quad g(z)=\sum_{k=0}^{\infty} d_k z^k$$

The Plancherel theorem and the modular 1 property of inner functions assert that
$$\sum_{k=0}^{\infty}\left|c_k\right|^2=|f|^2=|g|^2=\sum_{k=0}^{\infty}\left|d_k\right|^2$$
In digital signal processing (DSP) there is the following result: For any $n$,
$$\sum_{k=n}^{\infty}\left|c_k\right|^2 \geq \sum_{k=n}^{\infty}\left|d_k\right|^2$$
(see, for instance, $[11,19]$ ).
In DSP this is referred as energy-front-loading property of minimum phase signals. This amounts to saying that through factorizing out the inner function factor the convergence rate of the Fourier series of the remaining outer function becomes higher. This fact suggests that the AFD process would be better to incorporate with the factorization process for speeding up the convergence. This instructs that when a signal by its nature is of high frequency, one should first perform “unwinding” before extracting out from it a maximal portion of lower frequency. We proceed as follows [74, 92]. First we do factorization $f=f_1=I_1 O_1$, where $I_1$ and $O_1$ are, respectively, the inner and outer factors of $f$. The factorization is based on Nevanlinna’s factorization theorem, also see [117]. The outer function has the explicit integral representation
$$O_1(z)=e^{\frac{1}{2 \pi} \int_0^{2 \pi} \frac{e^{i t}+z}{e^{i t}-z} \log \left|f_1\left(e^{i t}\right)\right| d t}$$
The outer function is computed by using the boundary value of $f_1$. On the boundary the above integral is taken to be of the principal integral sense. The imaginary part of the integral reduces to the circular Hilbert transform of $\log \left|f_1\left(e^{i t}\right)\right|$. Next, we do a maximal sifting to $O_1$.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|One Dimensional Core-Adaptive Fourier Decomposition

$$e_a(z)=\frac{\sqrt{1-|a|^2}}{1-\bar{a} z}, \quad a \in \mathbf{D}$$

$$f(z)=\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)+\frac{f_1(z)-\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)}{\frac{z-a_1}{1-\bar{a}_1 z}} \frac{z-a_1}{1-\bar{a}_1 z}$$

## 数学代写|复分析作业代写Complex function代考|Unwinding AFD

$$f(z)=\sum_{k=0}^{\infty} c_k z^k, \quad g(z)=\sum_{k=0}^{\infty} d_k z^k$$
Plancherel 定理和内部函数的模 1 属性断言
$$\sum_{k=0}^{\infty}\left|c_k\right|^2=|f|^2=|g|^2=\sum_{k=0}^{\infty}\left|d_k\right|^2$$

$$\sum_{k=n}^{\infty}\left|c_k\right|^2 \geq \sum_{k=n}^{\infty}\left|d_k\right|^2$$
(例如，参见 $[11,19]$ ).

$$O_1(z)=e^{\frac{1}{2 \pi} \int_0^{2 \pi} \frac{e^{i t}+z}{e^{i t}-z} \log \left|f_1\left(e^{i t}\right)\right| d t}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|MAST30021

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Mergelyan Approximation in L2-Spaces

In his thesis from 2015, S. Gubkin [88] investigated Mergelyan approximation in $L^2$ spaces of holomorphic functions on pseudoconvex domains in $\mathbb{C}^n$ :
$$H^2(\Omega)=\mathscr{O}(\Omega) \cap L^2(\Omega) .$$
The following theorem generalizes both his main results [88, Theorems $4.2 .2$ and $4.3 .3]$; in the first one the domain is assumed to have $\mathscr{C}^{\infty}$-smooth boundary, and in the second one it is assumed to admit a $\mathscr{C}^2$ plurisubharmonic defining function. We only assume that the closure of the domain is a Stein compact.

Theorem 26 Assume that $X$ is a Stein manifold and $\Omega \Subset X$ is a relatively compact pseudoconvex domain with $\mathscr{C}^1$ boundary whose closure $\bar{\Omega}$ is a Stein compact. Then for any $f \in H^2(\Omega)$ there exists a sequence $f_j \in \mathscr{O}(\bar{\Omega})$ such that $\lim {j \rightarrow \infty} | f_j-$ $f |{L^2(\Omega)}=0$.

Proof As in the proof of Theorem 24, we find an open cover $\left{W_j\right}_{j=0}^l$ of $\bar{\Omega}{1 / m_0}$ for some $m_0 \in \mathbb{N}$ such that (22) holds. (This only requires that $b \Omega$ is of class $\mathscr{C}^1$.) Let $\left{\chi_j\right}{j=0}^l$ be a smooth partition of unity subordinate to $\left{W_j\right}_{j=0}^l$. Given an integer $m \geq m_0$ we define the cover $\left{U_{m, j}\right}_{j=0}^l$ and the functions $\left(f_{m, j}\right){j=0}^l$ by (23) and (24), respectively. Consider the function $$g_m=\sum{j=0}^l \chi_j f_{m, j} \in L^2\left(\Omega_{1 / m}\right)$$
Fix $\delta>0$. Since $|f|_{L^2(\Omega)}<\infty$, there exists a compact subset $K \subset \Omega$ such that
$$|f|_{L^2(\Omega \backslash K)}<\delta .$$
Choose a compact set $K^{\prime} \subset \Omega$ such that
$$K \cup \operatorname{supp}\left(\chi_0\right) \subset \stackrel{\circ}{K}^{\prime} .$$

## 数学代写|复分析作业代写Complex function代考|Carleman Approximation in Several Variables

Carleman approximation on the totally real affine subspace $M=\mathbb{R}^n \subset \mathbb{C}^n$ was proved by S. Scheinberg [147] in 1976. Such spaces are obviously polynomially convex, and, although less obviously so, they satisfy the following condition (compare with Definition 2). For any compact set $C \subset \mathbb{C}^n$ we set
$$h(C):=\overline{\widehat{C} \backslash C} .$$
Definition 6 A closed set $M \subset \mathbb{C}^n$ has the bounded exhaustion hulls property if for any polynomially convex compact set $K \subset \mathbb{C}^n$ there exists $R>0$ such that for any compact set $L \subset M$ we have that
$$h(K \cup L) \subset \mathbb{B}^n(0, R) .$$

Clearly, it suffices to test this condition on any increasing sequence of compact sets $K_j$ increasing to $\mathbb{C}^n$. This notion extends in an obvious way to closed sets in an arbitrary complex manifold $X$, replacing polynomial hulls by $\mathscr{O}(X)$-convex hulls. For closed sets $M$ in $\mathbb{C}$, this notion is equivalent to the one in Definition 2, and to the condition that $\mathbb{C P}^1 \backslash M$ is locally connected at infinity. (This is precisely the condition under which Arakelian’s Theorem 10 holds.)

To see that $M=\mathbb{R}^n$ has bounded exhaustion hulls in $\mathbb{C}^n$, we consider compact sets of the form
$$K_r=\left{z \in \mathbb{C}^n:\left|x_j\right| \leq r,\left|y_j\right| \leq r, j=1, \ldots n\right} .$$
Let us first look at a point $\tilde{z}=\tilde{x}+i \tilde{y} \in \mathbb{C}^n \backslash \mathbb{R}^n$ with $\left|\tilde{x}j\right|>(\sqrt{n}+1) r$ for some $j$. Consider the pluriharmonic polynomial $$f(z)=-\Re\left((z-\tilde{x})^2\right)=\sum{i=1}^n\left(y_i^2-\left(x_i-\tilde{x}_i\right)^2\right), \quad z \in \mathbb{C}^n .$$
A simple calculation shows that $f(z)<0$ holds for any point $z \in K_r$, and we clearly have $f \leq 0$ on $\mathbb{R}^n$ and $f(\tilde{z})=(\tilde{y})^2>0$. This shows that
$$h\left(K_r \cup \mathbb{R}^n\right) \subset\left{z \in \mathbb{C}^n:\left|x_j\right| \leq(\sqrt{n}+1) r, j=1, \ldots, n\right} .$$
Clearly we also have $h\left(K_r \cup \mathbb{R}^n\right) \subset\left{z \in \mathbb{C}^n:\left|y_j\right| \leq r, j=1, \ldots, n\right}$, and (29) follows.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Mergelyan Approximation in L2-Spaces

$$H^2(\Omega)=\mathscr{O}(\Omega) \cap L^2(\Omega) .$$

$$g_m=\sum j=0^l \chi_j f_{m, j} \in L^2\left(\Omega_{1 / m}\right)$$

$$K \cup \operatorname{supp}\left(\chi_0\right) \subset \stackrel{\circ}{K}$$

## 数学代写|复分析作业代写Complex function代考|Carleman Approximation in Several Variables

$$h(C):=\overline{\widehat{C} \backslash C}$$

$$h(K \cup L) \subset \mathbb{B}^n(0, R) .$$

$$f(z)=-\Re\left((z-\tilde{x})^2\right)=\sum i=1^n\left(y_i^2-\left(x_i-\tilde{x}_i\right)^2\right), \quad z \in \mathbb{C}^n$$

hhleft(K r \cup \mathbb ${R}^{\wedge} n \backslash$ ight) $\backslash$ subset $\backslash$ eft $\left{z \backslash\right.$ in $\backslash m a t h b b{C}^{\wedge} n$ :

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|MATH307

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Approximation on Totally Real Submanifolds

In this section we present an optimal $\mathscr{C}^k$-approximation result on totally real submanifolds. With essentially no extra effort we get approximation results on stratified totally real manifolds and on admissible sets (see Theorems 20 and 21).
There is a long history on approximation on totally real submanifolds, starting with J. Wermer [173] on curves and R. O. Wells [172] on real analytic manifolds. The first general result on approximation on totally real manifolds with various degrees of smoothness is due to L. Hörmander and J. Wermer [97]. Their work is based on $L^2$-methods for solving the $\bar{\partial}$-equation, and the passage from $L^2$ to $\mathscr{C}^k$-estimates led to a gap between the order $m$ of smoothness of the manifold $M$ on which the approximation takes place, and the order $k$ of the norm of the Banach space $\mathscr{C}^k(M)$ in which the approximation takes place. Subsequently, several authors worked on decreasing the gap between $m$ and $k$, introducing more precise integral kernel methods for solving $\bar{\partial}$. The optimal result with $m=k$ was eventually obtained by M. Range and Y.-T. Siu [139]. Subsequent improvements were made by F. Forstnerič, E. Løw, and N. Øvrelid [66] in 2001. They developed Henkin-type kernels adapted to this situation and obtained optimal results on approximation of $\bar{\partial}$-flat functions in tubes around totally real manifolds by holomorphic functions. In 2009 , B. Berndtsson [18] used $L^2$-theory to give a new approach to uniform approximation by holomorphic functions on compact zero sets of strongly plurisubharmonic functions. A novel byproduct of his method is that, in the case of polynomial approximation, one gets a bound on the degree of the approximating polynomial in terms of the closeness of the approximation.

We will not go into the details of the $L^2$ or the integral kernel approaches, but will instead present a method based on convolution with the Gaussian kernel which originates in the proof of Weierstrass’s Theorem 1 on approximating continuous functions on $\mathbb{R}$ by holomorphic polynomials. This approach is perhaps the most elementary one, and is particularly well suited for proving Runge-Mergelyan type approximation results with optimal regularity on (strongly) admissible sets. It seems that the first modern application of this method was made in 1981 by $\mathrm{S}$. Baouendi and F. Treves [12] to obtain local approximation of Cauchy-Riemann (CR) functions on CR submanifolds. The use of this method on totally real manifolds was developed further by P. Manne [118] in 1993 to obtain Carleman approximation on totally real submanifolds (see also [119]).

## 数学代写|复分析作业代写Complex function代考|Approximation on Strongly Pseudoconvex Domains

As we have seen, proofs of the Mergelyan theorem in one complex variable depend heavily on integral representations of holomorphic or $\bar{\partial}$-flat functions. The single most important reason why the one-dimensional proofs work so well is that the Cauchy-Green kernel (4) provides a solution to the inhomogeneous $\bar{\partial}$-equation which is uniformly bounded on all of $\mathbb{C}$ in terms of sup-norm of the data and the area of its support (see (6)). This allows uniform approximation of functions in $\mathscr{A}(K)$ on any compact set $K \subset \mathbb{C}$ with not too rough boundary by functions in $\mathscr{O}(K)$ (see Vitushkin’s Theorem 7). Nothing like that holds in several variables, and the question of uniform approximability is highly sensitive to the shape of the boundary even for smoothly bounded domains.

The idea of developing holomorphic integral kernels for domains in $\mathbb{C}^n$ with comparable properties to those of the Cauchy kernel in one variable was promoted by H. Grauert already around 1960; however, it took almost a decade to be realized. The first such constructions were given in 1969 by G. M. Henkin [92] and E. Ramírez de Arellano [138] for the class of strongly pseudoconvex domains. These kernels provide solution operators for the $\bar{\partial}$-equation which are bounded in the $\mathscr{C}^k$ norms and improve the regularity by $1 / 2$. We state here a special case of their results for $(0,1)$-forms, but in a more precise form which can be found in the works by I. Lieb and M. Range [112, Theorem 1], I. Lieb and J. Michel [111], and [62, Theorem 2.7.3]. A brief historical review of the kernel method is given in [66, pp. 392-393]. Given a domain $\Omega \subset \mathbb{C}^n$, we denote by $\mathscr{C}_{(0,1)}^k(\bar{\Omega})$ the space of all differential $(0,1)$-forms of class $\mathscr{C}^k$ on $\bar{\Omega}$.

Theorem 23 If $\Omega$ is a bounded strongly pseudoconvex Stein domain with boundary of class $\mathscr{C}^k$ for some $k \in{2,3, \ldots}$ in a complex manifold $X$, there exists a bounded linear operator $T: \mathscr{\zeta}{(0,1)}^0(\bar{\Omega}) \rightarrow \mathscr{L}^0(\bar{\Omega})$ satisfying the following properties: (i) If $f \in \mathscr{C}{0,1}^0(\bar{\Omega}) \cap \mathscr{C}{0,1}^1(\Omega)$ and $\bar{\partial} f=0$, then $\bar{\partial}(T f)=f$. (ii) If $f \in \mathscr{C}{0,1}^0(\bar{\Omega}) \cap \mathscr{C}{0,1}^r(\Omega)$ for some $r \in{1, \ldots, k}$ then $$|T f|{\mathscr{G} l, 1 / 2(\bar{\Omega})} \leq C_{l, \Omega}|f|_{\mathscr{C}{0,1}(\bar{\Omega})}, \quad l=0,1, \ldots, r .$$ Moreover, the constants $C{l, \Omega}$ may be chosen uniformly for all domains sufficiently $\mathscr{C}^k$ close to $\bar{\Omega}$.

# 复分析代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|KMA152

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|The Oka–Weil Theorem and Its Generalizations

The analogue of Runge’s theorem (see Theorems 2 and 4) on Stein manifolds and Stein spaces is the following theorem due to K. Oka [132] and A. Weil [171]. All complex spaces are assumed to be reduced.

Theorem 18 (The Oka-Weil Theorem) If $X$ is a Stein space and $K$ is a compact $\mathscr{O}(X)$-convex subset of $X$, then every holomorphic function in an open neighborhood of $K$ can be approximated uniformly on $K$ by functions in $\mathscr{O}(X)$.

Proof Two proofs of this result are available in the literature. The original one, due to $\mathrm{K}$. Oka and A. Weil, proceeds as follows. A compact $\mathscr{O}(X)$-convex subset $K$ in a Stein space $X$ admits a basis of open Stein neighborhoods of the form
$$P=\left{x \in X:\left|h_1(x)\right|<1, \ldots,\left|h_N(x)\right|<1\right}$$
with $h_1, \ldots, h_N \in \mathscr{O}(X)$. We may assume that the function $f \in \mathscr{O}(K)$ to be approximated is holomorphic on $P$. By adding more functions if necessary, we can ensure that the map $h=\left(h_1, \ldots, h_N\right): X \rightarrow \mathbb{C}^N$ embeds $P$ onto a closed complex subvariety $A=h(P)$ of the unit polydisc $\mathbb{D}^N \subset \mathbb{C}^N$. Hence, there is a function $g \in \mathscr{O}(A)$ such that $g \circ h=f$ on $P$. By the Oka-Cartan extension theorem [62, Corollary 2.6.3], $g$ extends to a holomorphic function $G$ on $\mathbb{D}^N$. Expanding $G$ into a power series and precomposing its Taylor polynomials by $h$ gives a sequence of holomorphic functions on $X$ converging to $f$ uniformly on $K$.

Another approach uses the method of L. Hörmander for solving the $\bar{\partial}$-equation with $L^2$-estimates (see $[94,96]$ ). We consider the case $X=\mathbb{C}^n$; the general case reduces to this one by standard methods of Oka-Cartan theory. Assume that $f$ is a holomorphic function in a neighborhood $U \subset \mathbb{C}^n$ of $K$. Choose a pair of neighborhoods $W \Subset V \Subset U$ of $K$ and a smooth function $\chi: \mathbb{C}^n \rightarrow[0,1]$ such that $\chi=1$ on $\bar{V}$ and $\operatorname{supp}(\chi) \subset U$. By choosing $W \supset K$ small enough, there is a nonnegative plurisubharmonic function $\rho \geq 0$ on $\mathbb{C}^n$ that vanishes on $W$ and satisfies $\rho \geq c>0$ on $U \backslash V$. Note that the smooth $(0,1)$-form
$$\alpha=\bar{\partial}(\chi f)=f \bar{\partial} \chi=\sum_{i=1}^n \alpha_i d \bar{z}_i$$
is supported in $U \backslash V$. Hörmander’s theory for the $\bar{\partial}$-complex (see [96, Theorem 4.4.2]) furnishes for any $t>0$ a smooth function $h_t$ on $\mathbb{C}^n$ satisfying $\bar{\partial} h_t=\alpha \quad$ and $\quad \int_{\mathbb{C}^n} \frac{\left|h_t\right|^2}{\left(1+|z|^2\right)^2} e^{-t \rho} d \lambda \leq \int_{\mathbb{C}^n} \sum_{i=1}^n\left|\alpha_i\right|^2 e^{-t \rho} d \lambda$

## 数学代写|复分析作业代写Complex function代考|Mergelyan’s Theorem in Higher Dimensions

As we have seen in Sections 2-4, the Mergelyan approximation theory in the complex plane and on Riemann surfaces was a highly developed subject around mid twentieth century. Around the same time, it became clear that the situation is much more complicated in higher dimensions. For example, in $1955 \mathrm{~J}$. Wermer [173] constructed an arc in $\mathbb{C}^3$ which fails to have the Mergelyan property. This suggests that, in several variables, one has to be much more restrictive about the sêts on which one considers Mergeelyan type anpproximation problems.

There are two lines of investigations in the literature: approximation on submanifolds of $\mathbb{C}^n$ of various degrees of smoothness and approximation on closures of bounded pseudoconvex domains. In neither category the problem is completely understood, and even with these restrictions, the situation is substantially more complicated than in dimension one. For example, R. Basener (1973), [14] (generalizing a result of B. Cole (1968), [39]) showed that Bishop’s peak point criterium does not suffice even for smooth polynomially convex submanifolds of $\mathbb{C}^n$. Even more surprisingly, it was shown by K. Diederich and J. E. Fornæss in 1976 [42] that there exist bounded pseudoconvex domains with smooth boundaries in $\mathbb{C}^2$ on which the Mergelyan property fails. The picture for curves is more complete; see G. Stolzenberg [153], H. Alexander [5], and P. Gauthier and E. Zeron [80].

In this section we outline the developments starting around the 1960s, give proofs in some detail in the cases of totally real manifolds and strongly pseudoconvex domains, and provide some new results on combinations of such sets.

Definition 4 Let $(X, J)$ be a complex manifold, and let $M \subset X$ be a $\mathscr{C}^1$ submanifold.
(a) $M$ is totally real at a point $p \in M$ if $T_p M \cap J T_p M={0}$. If $M$ is totally real at all points, we say that $M$ is a totally real submanifold of $X$.
(b) $M$ is a stratified totally real submanifold of $X$ if $M=\bigcup_{i=1}^l M_i$, with $M_i \subset M_{i+1}$ locally closed sers, such that $M_1$ and $M_{i+1} \backslash M_i$ are torally real submanifolds.

We now introduce suitable types of sets for Mergelyan approximation. The following notion is a generalization of the one for Riemann surfaces in Definition 3 . Recall that a compact set $S$ in a complex manifold $X$ is a Stein compact if $S$ admits a basis of open Stein neighborhoods in $X$.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|The Oka–Weil Theorem and Its Generalizations

$$\alpha=\bar{\partial}(\chi f)=f \bar{\partial} \chi=\sum_{i=1}^n \alpha_i d \bar{z}i$$ 支持 $U \backslash V$. 霍曼德的理论 $\bar{\partial}$-complex（见[96，定理 4.4.2]） 提供任何 $t>0$ 平滑函数 $h_t$ 在 $\mathbb{C}^n$ 令人满意 $\bar{\partial} h_t=\alpha \quad$ 和 $\quad \int{\mathbb{C}^n} \frac{\left|h_t\right|^2}{\left(1+|z|^2\right)^2} e^{-t \rho} d \lambda \leq \int_{\mathbb{C}^n} \sum_{i=1}^n\left|\alpha_i\right|^2 e^{-t \rho} d \lambda$

## 数学代写|复分析作业代写Complex function代考|Mergelyan’s Theorem in Higher Dimensions

(一种) $M$ 在某一点上是完全真实的 $p \in M$ 如果 $T_p M \cap J T_p M=0$. 如果 $M$ 在所有点上都是完全真实 的，我们说 $M$ 是一个完全真实的子流形 $X$.
(乙) $M$ 是一个分层的完全真实的子流形 $X$ 如果 $M=\bigcup_{i=1}^l M_i ＼mathrm{~ ， 和 ~} M_i \subset M_{i+1}$ 本地封闭的 sers， 这样 $M_1$ 和 $M_{i+1} \backslash M_i$ 是真正的子流形。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|MATH2242

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Power series

Until now we have not discussed any specific examples of functions of a complex variable. Of course, there are the standard functions that you probably encountered already in your undergraduate studies: polynomials, rational functions, $e^z$, the trigonometric functions, etc. But aside from these examples, it would be useful to have a general way to construct a large family of functions. Of course, there is such a way: power series, which-nonobviously-turn out to be essentially as general a family of functions as one could hope for.

To make things precise, a power series is a function of a complex variable $z$ that is defined by
$$f(z)=\sum_{n=0}^{\infty} a_n z^n$$
where $\left(a_n\right){n=0}^{\infty}$ is a sequence of complex numbers, or more generally by $$g(z)=f\left(z-z_0\right)=\sum{n=0}^{\infty} a_n\left(z-z_0\right)^n$$
where $\left(a_n\right)_{n=0}^{\infty}$ is again a a sequence and $z_0$ is some fixed complex number. These functions are defined wherever the respective series converge.

For which values of $z$ does this formula make sense? It is not hard to see that it converges absolutely precisely for $0 \leq|z|<R$, where the value of $R$ is given by
$$R=\left(\limsup {n \rightarrow \infty}\left|a_n\right|^{1 / n}\right)^{-1} .$$ $R$ is called the radius of convergence of the power series. Proof. Assume $00$, we have that $\left|a_n\right|<\left(\frac{1}{R}+\epsilon\right)^n$ if $n$ is large enough, and $R$ is the minimal number with that property. Let $z \in D_R(0)$. Since $|z|0$ chosen small enough. That implies that for $n>N$ (for some large enough $N$ as a function of $\epsilon$ ), $$\sum{n=N}^{\infty}\left|a_n z^n\right|<\sum_{n=N}^{\infty}\left[\left(\frac{1}{R}+\epsilon\right)|z|\right]^n$$
so the series is dominated by a convergent geometric series, and hence converges.

Conversely, if $|z|>R$, then, $|z|\left(\frac{1}{R}-\epsilon\right)>1$ for some small enough fixed $\epsilon>0$. Taking a subsequence $\left(a_{n_k}\right){k=1}^{\infty}$ for which $\left|a{n_k}\right|>\left(\frac{1}{R}-\epsilon\right)^{n_k}$ (guaranteed to exist by the definition of $R$ ), we see that
$$\sum_{n=0}^{\infty}\left|a_n z^n\right| \geq \sum_{k=1}^{\infty}\left[|z|\left(\frac{1}{R}-\epsilon\right)\right]^{n_k}=\infty$$
so the power series diverges.

## 数学代写|复分析作业代写Complex function代考|Contour integrals

We now introduce contour integrals, which are another fundamental building block of the theory.

Contour integrals, like many other types of integrals, take as input a function to be integrated and a “thing” (or “place”) over which the function is integrated. In the case of contour integrals, the “thing” is a contour, which is (for our current purposes at least) a kind of planar curve. We start by developing some terminology to discuss such objects. First, there is the notion of a parametrized curve, which is simply a continuous function $\gamma:[a, b] \rightarrow \mathbb{C}$. The value $\gamma(a)$ is called the starting point and $\gamma(b)$ is called the ending point. Two curves $\gamma_1:[a, b] \rightarrow \mathbb{C}, \gamma_2:[c, d] \rightarrow \mathbb{C}$ are called equivalent, which is denoted $\gamma_1 \sim \gamma_2$, if $\gamma_2(t)=\gamma_1(I(t))$ where $I:[c, d] \rightarrow[a, b]$ is a continuous, one-to-one, onto, increasing function. A “curve” is an equivalence class of parametrized curves with respect to this equivalence relation.

In practice, we will usually refer to parametrized curves as “curves”, which is the usual abuse of terminology that one sees in various places in mathematics, in which one blurs the distinction between equivalence classes and their members, remembering that various definitions, notation, and proof arguments need to “respect the equivalence” in the sense that they do not depend of the choice of member. (Meta-exercise: think of 2-3 other examples of this phenomenon.)

For our present context of developing the theory of complex analysis, we shall assume all our curves are piecewise continuously differentiable. More generally, one can assume them to be rectifiable, but we will not bother to develop that theory. There are yet more general contexts in which allowing curves to be merely continuous is beneficial (and indeed some of the ideas we will develop in a complex-analytic context can be carried over to that more general setting), but we will not pursue such distractions either.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Power series

$$f(z)=\sum_{n=0}^{\infty} a_n z^n$$

$$g(z)=f\left(z-z_0\right)=\sum n=0^{\infty} a_n\left(z-z_0\right)^n$$

$$\sum_{n=0}^{\infty}\left|a_n z^n\right| \geq \sum_{k=1}^{\infty}\left[|z|\left(\frac{1}{R}-\epsilon\right)\right]^{n_k}=\infty$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|MATH307

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Definition and basic meanings of analyticity

Definition 1 (analyticity). A function $f(z)$ of a complex variable is holomorphic (a.k.a. complex-differentiable, analytic ${ }^1$ ) at $z$ if the limit
$$f^{\prime}(z):=\lim _{h \rightarrow 0} \frac{f(z+h)-f(z)}{h}$$
exists. In this case we call $f^{\prime}(z)$ the derivative of $f$ at $z$.
In the case when $f^{\prime}(z) \neq 0$, the existence of the derivative has a geometric meaning: if we write the polar decomposition $f^{\prime}(z)=r e^{i \theta}$ of the derivative, then for points $w$ that are close to $z$, we will have the approximate equality
$$\frac{f(w)-f(z)}{w-z} \approx f^{\prime}(z)=r e^{i \theta},$$
or equivalently
$$f(w) \approx f(z)+r e^{i \theta}(w-z)+\text { [lower order terms] },$$
where “lower order terms” refers to a quantity that is much smaller in magnitude that $|w-z|$. Geometrically, this means that to compute $f(w)$, we start from $f(z)$, and move by a vector that results by taking the displacement vector $w-z$, rotating it by an angle of $\theta$, and then scaling it by a factor of $r$ (which corresponds to a magnification if $r>1$, a shrinking if $0<r<1$, or doing nothing if $r=1$ ). This idea can be summarized by the slogan:
“Analytic functions behave locally as a rotation composed with a scaling.”

## 数学代写|复分析作业代写Complex function代考|The Cauchy-Riemann equations

In addition to the geometric picture associated with the definition of the complex derivative, there is yet another quite different but also extremely useful way to think about analyticity, that provides a bridge between complex analysis and ordinary multivariate calculus. Remembering that complex numbers are veclors that have real and imayinary components, we can denote $z=x+i y$, where $x$ and $y$ will denote the real and imaginary parts of the complex number $z$, and $f=u+i v$, where $u$ and $v$ are real-valued functions of $z$ (or equivalently of $x$ and $y$ ) that return the real and imaginary parts, respectively, of $f$. Now, if $f$ is analytic at $z$ then
\begin{aligned} f^{\prime}(z) & =\lim {h \rightarrow 0} \frac{f(z+h)-f(z)}{h} \ & =\lim {h \rightarrow 0, h \in \mathbb{R}} \frac{u(x+h+i y)-u(x+i y)}{h}+i \frac{v(x+h+i y)-v(x+i y)}{h} \ & =\frac{\partial u}{\partial x}+i \frac{\partial v}{\partial x} . \end{aligned}
On the other hand also
\begin{aligned} f^{\prime}(z) & =\lim {h \rightarrow 0} \frac{f(z+h)-f(z)}{h} \ & =\lim {h \rightarrow 0, h \in i \mathbb{R}} \frac{u(x+h+i y)-u(x+i y)}{h}+i \frac{v(x+h+i y)-v(x+i y)}{h} \ & =\lim _{h \rightarrow 0, h \in \mathbb{R}} \frac{u(x+i y+i h)-u(x+i y)}{i h}+i \frac{v(x+i y+i h)-v(x+i y)}{i h} \ & =-i \frac{\partial u}{\partial y}-i \cdot i \frac{\partial v}{\partial y}=\frac{\partial v}{\partial y}-i \frac{\partial u}{\partial y} . \end{aligned}
Since these limits are equal, by equating their real and imaginary parts we get a famous system of coupled partial differential equations, the CauchyRiemann equations:
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y} .$$
We have proved that if $f$ is analytic at $z=x+i y$ then the components $u, v$ of $f$ satisfy the Cauchy-Riemann equations. Conversely, we now claim if $f=u+i v$ is continuously differentiable at $z=x+i y$ (in the sense that each of $u$ and $v$ is a continuously differentiable function of $x, y$ as defined in ordinary real analysis) and satisfies the Cauchy-Riemann equations there, $f$ is analytic at $z$.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Definition and basic meanings of analyticity

$$f^{\prime}(z):=\lim _{h \rightarrow 0} \frac{f(z+h)-f(z)}{h}$$

$$\frac{f(w)-f(z)}{w-z} \approx f^{\prime}(z)=r e^{i \theta}$$

$$f(w) \approx f(z)+r e^{i \theta}(w-z)+[\text { lower order terms }]$$

“解析函数在局部表现为由缩放组成的旋转。”

## 数学代写|复分析作业代写Complex function代考|The Cauchy-Riemann equations

$$f^{\prime}(z)=\lim h \rightarrow 0 \frac{f(z+h)-f(z)}{h} \quad=\lim h \rightarrow 0, h \in \mathbb{R} \frac{u(x+h+i y)-u(x+i y)}{h}+i \frac{v}{h}$$

$$f^{\prime}(z)=\lim h \rightarrow 0 \frac{f(z+h)-f(z)}{h} \quad=\lim h \rightarrow 0, h \in i \mathbb{R} \frac{u(x+h+i y)-u(x+i y)}{h}+i$$

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|KMA152

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|why study complex analysis

These notes are about complex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. While this may sound a bit specialized, there are (at least) two excellent reasons why all mathematicians should learn about complex analysis. First, it is, in my humble opinion, one of the most beautiful areas of mathematics. One way of putting it that has occurred to me is that complex analysis has a very high ratio of theorems to definitions (i.e., a very low “entropy”): you get a lot more as “output” than you put in as “input.”

The second reason is complex analysis has a large number of applications (in both the pure math and applied math senses of the word) to things that seem like they ought to have little to do with complex numbers. For example:

• Solving polynomial equations: historically, this was the motivation for introducing complex numbers by Cardano, who published the famous formula for solving cubic equations in 1543 , after learning of the solution found earlier by Scipione del Ferro. An important point to keep in mind is that Cardano’s formula sometimes requires taking operations in the complex plane as an intermediate step to get to the final answer, even when the cubic equation being solved has only real roots.

Example 1. Using Cardano’s formula, it can be found that the solutions to the cubic equation
$$z^3+6 z^2+9 z+3=0$$
are
\begin{aligned} & z_1=2 \cos (2 \pi / 9)-2, \ & z_2=2 \cos (8 \pi / 9)-2, \ & z_3=2 \sin (\pi / 18)-2 \end{aligned}

• Proving Stirling’s formula: $n ! \sim \sqrt{2 \pi n}(n / e)^n$. Here, $a_n \sim b_n$ is the standard “asymptotic to” relation, defined to mean $\lim _{n \rightarrow \infty} a_n / b_n=1$.
• Proving the prime number theorem: $\pi(n) \sim \frac{n}{\log n}$, where $\pi(n)$ denotes the number of primes less than or equal to $n$ (the prime-counting function).

## 数学代写|复分析作业代写Complex function代考|The fundamental theorem of algebra

One of the most famous theorems in complex analysis is the not-very-aptly named Fundamental Theorem of Algebra. This seems like a fitting place to start our journey into the theory.

Theorem 1 (The Fundamental Theorem of Algebra.). Every nonconstant polynomial $p(z)$ over the complex numbers has a root.

The fundamental theorem of algebra is a subtle result that has many beautiful proofs. I will show you three of them. Let me know if you see any “algebra”…
First proof: analytic proof. Let
$$p(z)=a_n z^n+a_{n-1} z^{n-1}+\ldots+a_0$$
be a polynomial of degree $n \geq 1$, and consider where $|p(z)|$ attains its infimum.
First, note that it can’t happen as $|z| \rightarrow \infty$, since
$$|p(z)|=|z|^n \cdot\left(\left|a_n+a_{n-1} z^{-1}+a_{n-2} z^{-2}+\ldots+a_0 z^{-n}\right|\right)$$
and in particular $\lim {|z| \rightarrow \infty} \frac{|p(z)|}{|z|^n}=\left|a_n\right|$, so for large $|z|$ it is guaranteed that $|p(z)| \geq|p(0)|=\left|a_0\right|$. Fixing some radius $R>0$ for which $|z|>R$ implies $|p(z)| \geq\left|a_0\right|$, we therefore have that $$m_0:=\inf {z \in \mathbb{C}}|p(z)|=\inf {|z| \leq R}|p(z)|=\min {|z| \leq R}|p(z)|=\left|p\left(z_0\right)\right|$$
where $z_0=\underset{|z| \leq R}{\arg \min }|p(z)|$, and the minimum exists because $p(z)$ is a continuous function on the disc $D_R(0)$.

Denote $w_0=p\left(z_0\right)$, so that $m_0=\left|w_0\right|$. We now claim that $m_0=0$. Assume by contradiction that it doesn’t, and examine the local behavior of $p(z)$ around $z_0$; more precisely, expanding $p(z)$ in powers of $z-z_0$ we can write
$$p(z)-w_0+\sum_{j=1}^n c_j\left(z-z_0\right)^j-w_0+c_k\left(z-z_0\right)^k+\ldots+c_n\left(z-z_0\right)^n$$

where $k$ is the minimal positive index for which $c_j \neq 0$. (Exercise: why can we expand $p(z)$ in this way?) Now imagine starting with $z=z_0$ and traveling away from $z_0$ in some direction $e^{i \theta}$. What happens to $p(z)$ ? Well, the expansion gives
$$p\left(z_0+r e^{i \theta}\right)=w_0+c_k r^k e^{i k \theta}+c_{k+1} r^{k+1} e^{i(k+1) \theta}+\ldots+c_n r^n e^{i n \theta}$$

## 数学代写|复分析作业代写Complex function代考|why study complex analysis

• 求解多项式方程: 从历史上看，这是卡尔达诺引入复数的动机，卡尔达诺在学习了 Scipione del Ferro 较早发现的解后，于 1543 年发表了蓍名的求解三次方程的公式。需要牢记的重要一点是，卡 尔达诺公式有时需要在复平面上进行运算作为获得最终答案的中间步骤，即使所求解的三次方程只有 实根也是如此。
例 1. 利用卡尔达诺公式，可以求出三次方程的解
$$z^3+6 z^2+9 z+3=0$$

$$z_1=2 \cos (2 \pi / 9)-2, \quad z_2=2 \cos (8 \pi / 9)-2, z_3=2 \sin (\pi / 18)-2$$
• 证明斯特林公式: $n$ ! $\sqrt{2 \pi n}(n / e)^n$. 这里， $a_n \sim b_n$ 是标准的“渐近”关系，定义为 $\lim _{n \rightarrow \infty} a_n / b_n=1$.
• 证明素数定理: $\pi(n) \sim \frac{n}{\log n}$ ，在哪里 $\pi(n)$ 表示小于等于的素数个数 $n$ (质数计数函数）。

## 数学代写|复分析作业代写Complex function代考|The fundamental theorem of algebra

$$p(z)=a_n z^n+a_{n-1} z^{n-1}+\ldots+a_0$$

$$|p(z)|=|z|^n \cdot\left(\left|a_n+a_{n-1} z^{-1}+a_{n-2} z^{-2}+\ldots+a_0 z^{-n}\right|\right)$$

$$m_0:=\inf z \in \mathbb{C}|p(z)|=\inf |z| \leq R|p(z)|=\min |z| \leq R|p(z)|=\left|p\left(z_0\right)\right|$$

$$p(z)-w_0+\sum_{j=1}^n c_j\left(z-z_0\right)^j-w_0+c_k\left(z-z_0\right)^k+\ldots+c_n\left(z-z_0\right)^n$$

$$p\left(z_0+r e^{i \theta}\right)=w_0+c_k r^k e^{i k \theta}+c_{k+1} r^{k+1} e^{i(k+1) \theta}+\ldots+c_n r^n e^{i n \theta}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复分析作业代写Complex function代考|MAST30021

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Related Studies and Applications

The AFD type expansions is in a great extent related to the Beurling-Lax shiftinvariant subspaces of the Hardy $H^2$ spaces. In the unit disc case,
$$H^2(\mathbf{D})=\overline{\operatorname{span}}\left{B_k\right}_{k=1}^{\infty} \oplus \phi H^2(\mathbf{D}),$$
where $\left{B_k\right}_{k=1}^{\infty}$ is the TM system generated by a sequence $\left{a_1, \cdots, a_n, \cdots\right}$, where multiples are counted, and $\phi$ is the Blaschke product with the zeros $\left{a_1, \cdots, a_n, \cdots\right}$ including the multiples. Note that when a Blaschke product $\phi$ having $a_k$ ‘s as all its zeros does not exist, corresponding to the condition
$$\sum_{k=1}^{\infty}\left(1-\left|a_k\right|\right)<\infty,$$
then the associated TM system is a basis. Although this has been well known over a long time, its relations with adaptive expansions, as far as what are aware by the author, have not been brought up. The fact that TM systems being Schauder systems was proved in [93]. The space decomposition relation (26) was extended to $H^p$ spaces, where $p \neq 2$ [80]. Relations between backward shift invariant subspaces and bandlimited functions and Bedrosian identity $[80,107]$ were studied. There are open questions on whether there exist adaptive and fast converging expansions by using TM systems for the cases $p \neq 2$, and for $p=2$ how far one can extend AFD (26) to higher dimensions. The study has a great room to be further developed.

## 数学代写|复分析作业代写Complex function代考|Extra-Strong Uncertainty Principle

The phase and frequency studies in mono-component function theory lay certain foundations in digital signal processing. In related studies what is called extra-strong uncertainty principle
$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left(\int_{-\infty}^{\infty}|t-\langle t\rangle | \phi(t)-\langle\omega\rangle||f(t)|^2 d t\right)^2$$
was recently established [22], where $f$ is a real-valued signal, $\sigma_t^2$ and $\sigma_\omega^2$ are the standard deviations with respect to the time and the Fourier frequency, and $\langle t\rangle$ and $\langle\omega\rangle$ are the corresponding means. A weaker uncertainty principle of the same type was previously given by L. Cohen
$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left.\left.\left|\int_{-\infty}^{\infty}(t-\langle t\rangle)(\phi(t)-\langle\omega\rangle)\right| f(t)\right|^2 d t\right|^2$$
[13]. We further extended the above result to multi-dimensional contexts [21-24, 26].

## 数学代写|复分析作业代写Complex function代考|Related Studies and Applications

AFD 类型展开在很大程度上与 Hardy 的 Beurling-Lax 位移不变子空间相关 $H^2$ 空间。在单元盘盒中， 数，并且 $\phi$ 是零点的 Blaschke 积 \left{a_1, Icdots, a_n, Icdots\right } } \text { 包括倍数。请注意，当 Blaschke 产品 } \phi \text { 有 } a _ { k } 的因为它的所有䨌都不存在，对应于条件
$$\sum_{k=1}^{\infty}\left(1-\left|a_k\right|\right)<\infty$$

## 数学代写|复分析作业代写Complex function代考|Extra-Strong Uncertainty Principle

$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left(\int_{-\infty}^{\infty}|t-\langle t\rangle| \phi(t)-\left.\langle\omega\rangle|| f(t)\right|^2 d t\right)^2$$

$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left.\left.\left|\int_{-\infty}^{\infty}(t-\langle t\rangle)(\phi(t)-\langle\omega\rangle)\right| f(t)\right|^2 d t\right|^2$$
[13]. 我们进一步将上述结果扩展到多维上下文 [21-24, 26]。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。