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# 数学代写|偏微分方程代考Partial Differential Equations代写|Relation of the Logarithmic Potential to the Theory of Functions

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Relation of the Logarithmic Potential to the Theory of Functions

There is a close connection between the theory of two-dimensional harmonic functions and the theory of analytic functions of a complex variable. The class of analytic functions of a complex variable $z=x+i y$ consists of the complex functions of $z$ which possess a derivative at each point. It can be shown ${ }^1$ that if $\phi$ and $\psi$ are the real and imaginary parts of an analytic function of the complex variable $x+i y$, then $\phi$ and $\psi$ must satisfy the Cauchy-Riemann equations
$$\frac{\partial \phi}{\partial x}=\frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y}=-\frac{\partial \psi}{\partial x}$$
Now it can be proved that the derivative of an analytic function is itself analytic, so that the functions $\phi$ and $\psi$ will have continuous partial derivatives of all orders and, in particular, Schwartz’s theorem
$$\frac{\partial^2 \phi}{\partial x \partial y}=\frac{\partial^2 \phi}{\partial y \partial x}, \quad \frac{\partial^2 \psi}{\partial x \partial y}=\frac{\partial^2 \psi}{\partial y \partial x}$$
will hold. Combining the results (1) and (2), we then find that
$$\nabla_1^2 \phi=\nabla_1^2 \psi=0$$
i.e., the real and imaginary parts of an analytic function are harmonic functions. The functions $\phi, \psi$ so defined are called conjugate functions.
The converse result is also true: If the harmonic functions $\phi$ and $\psi$ satisfy the Cauchy-Riemann equations, then $\phi+i \psi$ is an analytic function of $z=x+i y$.

If either $\phi(x, y)$ or $\psi(x, y)$ is given, it is possible to determine the analytic function $w=\phi+i \psi$, for, by equations (1),
$$\frac{d w}{d z}=\frac{\partial \phi}{\partial x}+i \frac{\partial \psi}{\partial x}=\phi_1(x, y)-i \phi_2(x, y)$$
where $\phi_1=\partial \phi / \partial x, \phi_2=\partial \phi / \partial y$. Putting $y=0$, we have the identity
$$\frac{d w}{d z}=\phi_1(z, 0)-i \phi_2(z, 0)$$
from which $w$ may be derived by a simple integration. If $\psi$ is given, then, in a similar notation,
$$\frac{d w}{d z}=\psi_2(z, 0)+i \psi_1(z, 0)$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Green’s Function for the Two-dimensional Equation

The theory of the Green’s function for the two-dimensional Laplace equation may be developed along lines similar to those of Sec. 8 . If we put
$$P=-\psi \frac{\partial \psi^{\prime}}{\partial y}, \quad Q=\psi \frac{\partial \psi^{\prime}}{\partial x}$$
in equation (4) of Sec. 11 , we find that
$$\int_K \psi \nabla_1^2 \psi^{\prime} d s+\int_K\left(\frac{\partial \psi}{\partial x} \frac{\partial \psi^{\prime}}{\partial x}+\frac{\partial \psi}{\partial y} \frac{\partial \psi^{\prime}}{\partial y}\right) d S=\int_C \psi \frac{\partial \psi^{\prime}}{\partial n} d S$$
If we interchange $\psi$ and $\psi^{\prime}$ and subtract the two equations, we find that
$$\int_R\left(\psi \nabla_1^2 \psi^{\prime}-\psi^{\prime} \nabla_1^2 \psi\right) d S=\int_C\left(\psi \frac{\partial \psi^{\prime}}{\partial n}-\psi^{\prime} \frac{\partial \psi}{\partial n}\right) d S$$
Suppose that $P$ with coordinates $(x, y)$ is a point in the interior of the region $S$ in which the function $\psi$ is assumed to be harmonic. Draw a circle. $\Gamma$ with center $P$ and small radius $\varepsilon$ (cf. Fig. 33), and apply the result (2) to the region $K$ bounded by the curves $C$ and $\Gamma$ with
$$\psi^{\prime}=\log \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$$
Since both $\psi$ and $\psi^{\prime}$ are harmonic, it follows that if $s$ is measured in the directions shown in Fig. 33,
$$\left(\int_{\Gamma}+\int_C\right)\left{\psi\left(x^{\prime}, y^{\prime}\right) \frac{\partial}{\partial n} \log \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}-\log \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \frac{\partial \psi}{\partial n}\right} d s^{\prime}=0$$

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Relation of the Logarithmic Potential to the Theory of Functions

$$\frac{\partial \phi}{\partial x}=\frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y}=-\frac{\partial \psi}{\partial x}$$

$$\frac{\partial^2 \phi}{\partial x \partial y}=\frac{\partial^2 \phi}{\partial y \partial x}, \quad \frac{\partial^2 \psi}{\partial x \partial y}=\frac{\partial^2 \psi}{\partial y \partial x}$$

$$\nabla_1^2 \phi=\nabla_1^2 \psi=0$$

$$\frac{d w}{d z}=\frac{\partial \phi}{\partial x}+i \frac{\partial \psi}{\partial x}=\phi_1(x, y)-i \phi_2(x, y)$$

$$\frac{d w}{d z}=\phi_1(z, 0)-i \phi_2(z, 0)$$

$$\frac{d w}{d z}=\psi_2(z, 0)+i \psi_1(z, 0)$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Green’s Function for the Two-dimensional Equation

$$P=-\psi \frac{\partial \psi^{\prime}}{\partial y}, \quad Q=\psi \frac{\partial \psi^{\prime}}{\partial x}$$

$$\int_K \psi \nabla_1^2 \psi^{\prime} d s+\int_K\left(\frac{\partial \psi}{\partial x} \frac{\partial \psi^{\prime}}{\partial x}+\frac{\partial \psi}{\partial y} \frac{\partial \psi^{\prime}}{\partial y}\right) d S=\int_C \psi \frac{\partial \psi^{\prime}}{\partial n} d S$$

$$\int_R\left(\psi \nabla_1^2 \psi^{\prime}-\psi^{\prime} \nabla_1^2 \psi\right) d S=\int_C\left(\psi \frac{\partial \psi^{\prime}}{\partial n}-\psi^{\prime} \frac{\partial \psi}{\partial n}\right) d S$$

$$\psi^{\prime}=\log \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$$

$$\left(\int_{\Gamma}+\int_C\right)\left{\psi\left(x^{\prime}, y^{\prime}\right) \frac{\partial}{\partial n} \log \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}-\log \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \frac{\partial \psi}{\partial n}\right} d s^{\prime}=0$$以上翻译结果来自有道神经网络翻译（YNMT）· 通用场景

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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。