### 数学代写|matlab代写|CS1132

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

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• 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|matlab代写|THE CAUCHY-RIEMANN EQUATIONS

In the previous two sections, we introduced complex arithmetic. We are now ready for the concept of function as it applies to complex variables.

We already defined the complex variable $z=x+i y$, where $x$ and $y$ are variable. We now introduce another complex variable $w-u+i v$ so that for each value of $z$ there corresponds a value of $w=f(z)$. From all of the possible complex functions that we might invent, we focus on those functions where for each $z$ there is one, and only one, value of $w$. These functions are single-valued. They differ from functions such as the square root, logarithm, and inverse sine and cosine, where there are multiple answers for each $z$. These multivalued functions do arise in various problems. However, they are beyond the scope of this book and we shall always assume that we are dealing with single-valued functions.

A popular method for representing a complex function involves drawing some closed domain in the $z$-plane and then showing the corresponding domain in the $w$-plane. This procedure is called mapping and the $z$-plane illustrates the domain of the function while the $w$-plane illustrates its image or rangc. Figure 1.3.1 shows the $z$-plane and $w$-plane for $w=z^{2}$; a pie-shaped wedge in the $z$-plane maps into a semicircle on the $w$-plane.

## 数学代写|matlab代写|LINE INTEGRALS

So far, we discussed complex numbers, complex functions, and complex differentiation. We are now ready for integration.

Just as we have integrals involving real variables, we can define an integral that involves complex variables. Because the $z$-plane is two-dimensional, there is clearly greater freedom in what we mean by a complex integral. For example, we might ask whether the integral of some function between points $A$ and $B$ depends upon the curve along which we integrate. (In general it does.) Consequently, an important ingredient in any complex integration is the contour that we follow during the integration.

The result of a line integral is a complex number or expression. Unlike its counterpart in real variables, there is no physical interpretation for this quantity, such as area under a curve. Generally, integration in the complex plane is an intermediate process with a physically realizable quantity occurring only after we take its real or imaginary part. For example, in potential fluid flow, the lift and drag are found by taking the real and imaginary parts of a complex integral, respectively.

How do we compute $\int_{C} f(z) d z$ ? Let us deal with the definition; we illustrate the actual method by examples.

A popular method for evaluating complex line integrals consists of breaking everything up into real and imaginary parts. This reduces the integral to line integrals of real-valued functions, which we know how to handle. Thus, we write $f(z)=u(x, y)+i v(x, y)$ as usual, and because $z=x+i y$, formally $d z=d x+i d y$. Therefore,
\begin{aligned} \int_{C} f(z) d z &=\int_{C}[u(x, y)+i v(x, y)][d x+i d y] \ &=\int_{C} u(x, y) d x-v(x, y) d y+i \int_{C} v(x, y) d x+u(x, y) d y . \end{aligned}

## 数学代写|matlab代写|LINE INTEGRALS

$$\int_{C} f(z) d z=\int_{C}[u(x, y)+i v(x, y)][d x+i d y]=\int_{C} u(x, y) d x-v(x, y) d y+i \int_{C} v(x, y) d x+u(x$$

## 有限元方法代写

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## 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。