### 数学代写|复变函数作业代写Complex function代考|Complex Line Integrals

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• Foundations of Data Science 数据科学基础

## 数学代写|复变函数作业代写Complex function代考|Real and Complex Line Integrals

In the previous chapter, we approached the question of finding a function with given partial derivatives by integrating along vertical and horizontal directions only. The fact that the horizontal derivative is $\partial / \partial x$ and the vertical derivative is $\partial / \partial y$ then made the computations in Section $1.5$ obvious. But the restriction to such integrals is geometrically unnatural. In this section we are going to develop an integration process along more general curves. It is in fact not a new method of integration at all but is the process of line integration which you learned in calculus. Our chief job here is to make it rigorous and to introduce notation that is convenient for complex analysis.

First, let us define the class of curves we shall consider. It is convenient to think of a curve as a (continuous) function $\gamma$ from a closed interval $[a, b] \subseteq \mathbb{R}$ into $\mathbb{R}^{2} \approx \mathbb{C}$. Although it is frequently convenient to refer to the geometrical object $\tilde{\gamma} \equiv{\gamma(t): t \in[a, b]}$, most of our analysis will be done with the function $\gamma$. It is often useful to write
$$\gamma(t)=\left(\gamma_{1}(t), \gamma_{2}(t)\right) \quad \text { or } \quad \gamma(t)=\gamma_{1}(t)+i \gamma_{2}(t),$$
depending on the context. The curve $\gamma$ is called closed if $\gamma(a)=\gamma(b)$. It is called simple closed if $\left.\gamma\right|_{[a, b)}$ is one-to-one and $\gamma(a)=\gamma(b)$. Intuitively, a simple closed curve is a curve with no self-intersections, except of course for the closing up at $t=a, t=b$.

In order to work effectively with $\gamma$, we need to impose on it some differentiability properties. Since $\gamma$ is defined on a closed interval, this requires a new definition.

Definition 2.1.1. A function $\phi:[a, b] \rightarrow \mathbb{R}$ is called continuously differentiable (or $\left.C^{1}\right)$, and we write $\phi \in C^{1}([a, b])$, if
(a) $\phi$ is continuous on $[a, b]$
(b) $\phi^{\prime}$ exists on $(a, b)$;
(c) $\phi^{\prime}$ has a continuous extension to $[a, b]$.
In other words, we require that
$$\lim {t \rightarrow a^{+}} \phi^{\prime}(t) \text { and } \lim {t \rightarrow b^{-}} \phi^{\prime}(t)$$
both exist.
The motivation for the definition is that if $\phi \in C^{1}([a, b])$ and $\phi$ is realvalued, then
\begin{aligned} \phi(b)-\phi(a) &=\lim {\epsilon \rightarrow 0^{+}}(\phi(b-\epsilon)-\phi(a+\epsilon)) \ &=\lim {\epsilon \rightarrow 0^{+}} \int_{a+\epsilon}^{b-\epsilon} \phi^{\prime}(t) d t \ &=\int_{a}^{b} \phi^{\prime}(t) d t \end{aligned}
So the fundamental theorem of calculus holds for $\phi \in C^{1}([a, b])$.

## 数学代写|复变函数作业代写Complex function代考|Complex Differentiability and Conformality

The goal of our work so far has been to develop a complex differential and integral calculus. We recall from ordinary calculus that when we differentiate functions on $\mathbb{R}^{2}$, we consider partial derivatives and directional derivatives, as well as a total derivative. It is a very nice byproduct of the field structure of $\mathbb{C}$ that we may now unify these ideas in the complex case.

First we need a suitable notion of limit. The definition is in complete analogy with the usual definition in calculus:

Let $U \subseteq \mathbb{C}$ be open, $P \in U$, and $g: U \backslash{P} \rightarrow \mathbb{C}$ a function. We say that
$$\lim _{z \rightarrow P} g(z)=\ell, \quad \ell \in \mathbb{C},$$
if for any $\epsilon>0$ there is a $\delta>0$ such that when $z \in U$ and $0<|z-P|<\delta$, then $|g(z)-\ell|<\epsilon$.

In a similar fashion, if $f$ is a complex-valued function on an open set $U$ and $P \in U$, then we say that $f$ is continuous at $P$ if $\lim {z \rightarrow P} f(z)=f(P)$. Now let $f$ be a function on the open set $U$ in $\mathbb{C}$ and consider, in analogy with one variable calculus, the difference quotient $$\frac{f(z)-f\left(z{0}\right)}{z-z_{0}}$$
for $z_{0} \neq z \in U$. In case
$$\lim {z \rightarrow z{0}} \frac{f(z)-f\left(z_{0}\right)}{z-z_{0}}$$
exists, then we say that $f$ has a complex derivative at $z_{0}$. We denote the complex derivative by $f^{\prime}\left(z_{0}\right)$. Observe that if $f$ has a complex derivative at $z_{0}$, then certainly $f$ is continuous at $z_{0}$.

The classical method of studying complex function theory is by means of the complex derivative. We take this opportunity to tie up the (well motivated) classical viewpoint with our present one.

## 数学代写|复变函数作业代写Complex function代考|Antiderivatives Revisited

It is our goal in this section to extend Theorems 1.5.1 and $1.5 .3$ to the situation where $f$ and $g$ (for Theorem 1.5.1) and $F$ (for Theorem 1.5.3) have isolated singularities. These rather technical results will be needed for our derivation of what is known as the Cauchy integral formula in the next section. In particular, we shall want to study the complex line integral of
$$\frac{F(z)-F\left(z_{0}\right)}{z-z_{0}}$$

when $F$ is holomorphic on $U$ and $z_{0} \in U$ is fixed. Such a function is certainly $C^{1}$ on $U \backslash\left{z_{0}\right}$. But it is a priori known only to be continuous on the entire set $U$ (if it is defined to equal $F^{\prime}\left(z_{0}\right)$ at $z_{0}$ ). We need to deal with this situation, and doing so is the motivation for the rather technical refinements of this section.
We begin with a lemma about functions on $\mathbb{R}$.
Lemma 2.3.1. Let $(\alpha, \beta) \subseteq \mathbb{R}$ be an open interval and let $H:(\alpha, \beta) \rightarrow \mathbb{R}$, $F:(\alpha, \beta) \rightarrow \mathbb{R}$ be continuous functions. Let $p \in(\alpha, \beta)$ and suppose that $d H / d x$ exists and equals $F(x)$ for all $x \in(\alpha, \beta) \backslash{p}$. See Figure 2.2. Then $(d H / d x)(p)$ exists and $(d H / d x)(x)=F(x)$ for all $x \in(\alpha, \beta)$.

Proof. It is enough to prove the result on a compact subinterval $[a, b]$ of $(\alpha, \beta)$ that contains $p$ in its interior. Set
$$K(x)=H(a)+\int_{a}^{x} F(t) d t$$
Then $K^{\prime}(x)$ exists on all of $[a, b]$ and $K^{\prime}(x)=F(x)$ on both $[a, p)$ and $(p, b]$. Thus $K$ and $H$ differ by constants on each of these half open intervals. Since both functions are continuous on all of $[a, b]$, it follows that $K-H$ is constant on all of $[a, b]$. Since $(K-H)(a)=0$, it follows that $K \equiv H$.

Theorem 2.3.2. Let $U \subseteq \mathbb{C}$ be either an open rectangle or an open disc and let $P \in U$. Let $f$ and $g$ be continuous, real-valued functions on $U$ which are continuously differentiable on $U \backslash{P}$ (note that no differentiability hypothesis is made at the point $P$ ). Suppose further that
$$\frac{\partial f}{\partial y}=\frac{\partial g}{\partial x} \quad \text { on } U \backslash{P}$$
Then there exists a $C^{1}$ function $h: U \rightarrow \mathbb{R}$ such that
$$\frac{\partial h}{\partial x}=f, \quad \frac{\partial h}{\partial y}=g$$
at every point of $U$ (including the point $P$ ).
Proof. As in the proof of Theorem 1.5.1, we fix a point $\left(a_{0}, b_{0}\right)=a_{0}+i b_{0} \in$ $U$ and define
$$h(x, y)=\int_{a_{0}}^{x} f\left(t, b_{0}\right) d t+\int_{b_{0}}^{y} g(x, s) d s$$

## 数学代写|复变函数作业代写Complex function代考|Real and Complex Line Integrals

C(吨)=(C1(吨),C2(吨)) 或者 C(吨)=C1(吨)+一世C2(吨),

(a)φ是连续的[一种,b]
(二)φ′存在于(一种,b);
（C）φ′有一个连续的延伸到[一种,b].

φ(b)−φ(一种)=林ε→0+(φ(b−ε)−φ(一种+ε)) =林ε→0+∫一种+εb−εφ′(吨)d吨 =∫一种bφ′(吨)d吨

## 数学代写|复变函数作业代写Complex function代考|Antiderivatives Revisited

F(和)−F(和0)和−和0

ķ(X)=H(一种)+∫一种XF(吨)d吨

∂F∂是=∂G∂X 在 在∖磷

∂H∂X=F,∂H∂是=G

H(X,是)=∫一种0XF(吨,b0)d吨+∫b0是G(X,s)ds

## 有限元方法代写

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## MATLAB代写

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