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• 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|Harmonic Functions

Let $F$ be a holomorphic function on an open set $U \subseteq \mathbb{C}$. Write $F=u+i v$, where $u$ and $v$ are real-valued. We have already observed that the real part $u$ satisfies a certain partial differential equation known as Laplace’s equation:
$$\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) u=0 .$$
(Of course the imaginary part $v$ satisfies the same equation.) In this chapter we shall examine systematically those $C^{2}$ functions that satisfy this equation. They are called harmonic functions.

There are two main justifications for this study: First, the consideration of harmonic functions illuminates the behavior of holomorphic functions: If two holomorphic functions on a connected open set have the same real part, then the difference of the functions attains only imaginary values and, by the open mapping theorem, the difference is therefore an imaginary constant. Thus the real part (or the imaginary part) of a holomorphic function already contains essentially complete information about the holomorphic function itself. Second, harmonic functions are the most classical and fundamental instance of solutions of what are known as linear, elliptic partial differential equations. The detailed consideration of any general results from the theory of partial differential equations far exceeds the scope of this text. But it is important that harmonic function theory fits into this larger context (see [KRA2] for more on these matters). One of the most fascinating features of basic complex function theory is that it is the birthplace of a number of other branches of mathematics. The theory of elliptic partial differential equations is one of these.

数学代写|复分析作业代写Complex function代考|Basic Properties of Harmonic Functions

Recall from Section $1.4$ the precise definition of harmonic function:
Definition 7.1.1. A real-valued function $u: U \rightarrow \mathbb{R}$ on an open set $U \subseteq \mathbb{C}$ is harmonic if it is $C^{2}$ on $U$ and
$$\Delta u \equiv 0,$$
where the Laplacian $\Delta u$ is defined by
$$\Delta u=\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) u .$$
This definition applies as well to complex-valued functions. A complexvalued function is harmonic if and only if its real and imaginary parts are each harmonic (Exercise 16). There is hardly any reason, at least iny this text, to consider complex-valued har nic functions in their own right; we seldom do so.

The first thing that we need to check is that real-valued harmonic functions really are just those functions that arise as the real parts of holomorphic functions-at least locally. (We shall see later that certain complications arise when the harmonic function is defined on a set that is not holomorphically simply connected; for now we confine ourselves to the disc.)
Lemma 7.1.2. If $u: \mathcal{D} \rightarrow \mathbb{R}$ is a harmonic function on a disc $\mathcal{D}$, then there is a holomorphic function $F: \mathcal{D} \rightarrow \mathbb{C}$ such that $\operatorname{Re} F \equiv u$ on $\mathcal{D}$.

Proof. (Corollary 1.5.2. For convenience, we recall the proof.) We want to find a $v: \mathcal{D} \rightarrow \mathbb{R}$ such that
$$u+i v: \mathcal{D} \rightarrow \mathbb{C}$$
is holomorphic. Note that a $C^{1}$ function $v: \mathcal{D} \rightarrow \mathbb{R}$ will make $u+i v$ holomorphic if and only if
$$\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y} \quad \text { and } \quad \frac{\partial v}{\partial y}=\frac{\partial u}{\partial x} .$$

复分析代写

数学代写|复分析作业代写Complex function代考|Harmonic Functions

$$\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) u=0$$
(当然是虚部 $v$ 满足相同的方程。) 在本章中，我们将系统地研究那些 $C^{2}$ 满足这个方程的函数。它们被称为调和函数。

数学代写|复分析作业代写Complex function代考|Basic Properties of Harmonic Functions

$$\Delta u \equiv 0,$$

$$\Delta u=\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) u$$

$$u+i v: \mathcal{D} \rightarrow \mathbb{C}$$

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

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