### 数学代写|微积分代写Calculus代写|MAST10006

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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|微积分代写Calculus代写|Analytic Functions and the Derivative

The earliest signs that complex analysis would grow into an important mathematical study appeared during the sixteenth century Italian Renaissance, more than 70 years before Descartes invented the plane now named after him to explain Greek geometry in terms of equations, and more than 100 years before Newton invented calculus. In 1545 , Girolamo Cardano published the Ars Magna, a book that explained the rules of algebra. $^{1}$ In its Chapter XI, Cardano presented an equation that gave the zeros of any cubic polynomial. It worked the same way that the quadratic equation gave the zeros of any quadratic. The equation used a keen unpublished formula independently discovered thirty years earlier by other Italians, Niccoló Fontana (better known as Tartaglia_”the stammerer”) and Sciopione del Ferro. Just as with the quadratic equation, Tartaglia’s (and then Cardano’s) cubic formula contained radicals, and some cubic polynomials produced square roots of negative numbers. (Notes at the end of the chapter give further details.) Cardano even rolled out one such example. The cubic $x^{3}-15 x-4=0$ caused Tartaglia’s formula to spit out the term
$$x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$$
As had been the practice since ancient times when using the quadratic equation, Cardano dismissed these cases. Tartaglia’s formula didn’t always work. After all, how could the square root of a negative number make sense? Surely square roots of negative numbers could never be useful!

A young, informally trained mathematician Raphael Bombelli then entered the story in an incredibly powerful way. Around age 35 , he decided to dedicate the rest of his life to writing what he considered an important mathematics text in a sequence of volumes. Titled L’Algebra and published in 1572, it expressed known mathematics, especially the Italian algebra formulated in and around his hometown Bologna, in a less technical manner with less jargon so that a wider audience could appreciate it and use it. He also wrote about new ideas-ones that he alone understood.

## 数学代写|微积分代写Calculus代写|The Complex Derivative

A complex function $f$ has the same basic structure as does a real function, except it is not restricted to real numbers and instead can act on complex numbers and produce a complex range. In symbols, $f$ maps every (complex number) domain input $z=x+i y$ to a unique (complex number) output $f(z)=u+i v$. With two input variables $x$ and $y$ and two output variables $u$ and $v$, you can see that the graph of $f(z)$ would generally have to be displayed in four-dimensional space, and so visual displays of complex functions are very difficult. ${ }^{2}$ We call $x$ the real part of $z$ and $y$ the imaginary part and write
$$x=\operatorname{Re}[z] \text { and } y=\operatorname{Im}[z] .$$
In the same way
$$u=\operatorname{Re}[f(z)] \text { and } v=\operatorname{Im}[f(z)] .$$
This book studies complex functions. It is important to see how algebraic operations on complex numbers can produce functions. The simplest operations are addition, subtraction, multiplication, and division. Any combination of them as they act on a variable $z$ always produces a function. For example, the function
$$f(z)=z+(2+3 i)$$
uses the algebraic operation of addition of complex numbers. It has
$$f(4+5 t)=(4+5 t)+(2+3 t)=6+8 t$$
because addition is defined in terms of the sums of the individual real and imaginary components:
$$(a+i b)+(c+i d)=(a+c)+i(b+d)$$

## 数学代写|微积分代写Calculus代写|Analytic Functions and the Derivative

$$x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$$

## 数学代写|微积分代写Calculus代写|The Complex Derivative

$$x=\operatorname{Re}[z] \text { and } y=\operatorname{Im}[z]$$

$$u=\operatorname{Re}[f(z)] \text { and } v=\operatorname{Im}[f(z)]$$

$$f(z)=z+(2+3 i)$$

$$f(4+5 t)=(4+5 t)+(2+3 t)=6+8 t$$

$$(a+i b)+(c+i d)=(a+c)+i(b+d)$$

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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。