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# 数学代写|复分析代写Complex analysis代考|Regular Paths and Curves

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## 数学代写|复分析代写Complex analysis代考|Regular Paths and Curves

Just as we distinguish between a path $\gamma$ and its image curve, we must distinguish between the derivative $\gamma^{\prime}(t)$ and a tangent line to the curve. The derivative can be interpreted as the velocity vector at time $t$ for a point $\gamma(t)$ moving along the curve. If $\gamma^{\prime}(t) \neq 0$ it defines a tangent direction, hence a tangent line to the curve. When $\gamma^{\prime}(t)=0$ it does not defines a tangent direction, so the curve may not have a tangent line. Section 6.7 shows some of the things that can then happen.
First, some standard terminology:
DEFINITION 6.20. Let $\gamma:[a, b] \rightarrow \mathbb{C}$ be a smooth path.
If $t_0 \in[a, b]$ and $\gamma^{\prime}\left(t_0\right) \neq 0$, then $t_0$ is a regular point of $\gamma$.
If $t_0 \in[a, b]$ and $\gamma^{\prime}\left(t_0\right)=0$, then $t_0$ is a singular point of $\gamma$.
When the image curve has a well-defined tangent line, it looks smooth: see Proposition 6.22 below.

The above discussion leads naturally to a special type of path or curve that will be useful as we proceed, to relate the abstract theory to geometric intuition:

DEFINITION 6.21. A regular path is a smooth path $\gamma:[a, c] \rightarrow \mathbb{C}$ such that $\gamma^{\prime}(t) \neq 0$ for all $t \in[a, b]$.
That is, every point on the path is a regular point.
A regular curve is the image of a regular path.
If $\gamma$ is regular, then by Proposition 4.18 a point on the tangent at $\gamma(t)$ is of the form $\gamma(t)+h \gamma^{\prime}(t)$ for any $h \in \mathbb{R}$, Figure 6.6.
The standard paths $L(t)$ (line) and $C(t)$ (circle) in Section 2.4 are regular.
In Figure 6.6 the tangent line at $\gamma(t)$ is a good approximation to the curve given by the image of $\gamma$, near that point. To formalise this idea, we compare the path $\gamma(t)$ for $t$ near some point $t_0 \in[a, b]$ with the corresponding tangent line. We can think of the tangent line as a path $\tau$ in its own right, defined by
$$\tau\left(t_0+h\right)=\gamma\left(t_0\right)+h \gamma^{\prime}\left(t_0\right) \quad(h \in \mathbb{R})$$
and compare it with
$$\gamma\left(t_0+h\right) \quad\left(t \text { near } t_0\right)$$

## 数学代写|复分析代写Complex analysis代考|Parametrisation by Arc Length

Proposition 6.22 is a formal statement of the intuitive idea that a regular curve looks smooth near any point. It has a continuously turning tangent and a well-defined finite length. These properties are inherited by subpaths, leading to:

DEFINITION 6.23. Let $\gamma:[a, b] \rightarrow \mathbb{C}$ be a regular path, with image curve $C$. Let $t_0, t_1 \in[a, b]$ with $t_0 \leq t_1$, and let $\gamma\left(t_0\right)=c, \gamma\left(t_1\right)=d$. Then the arc length $L_C(c, d)$ from $c$ to $d$ in $C$ is the length of $\left.\gamma\right|{\left[t_0, t_1\right]}$; that is, $$L_C(c, d)=\int{t_0}^{t_1}\left|\gamma^{\prime}(t)\right| \mathrm{d} t$$
We now prove that a regular curve can be smoothly reparametrised so that the parameter $t$ is arc length, or a constant multiple of arc length if that is more convenient. Let the length of $\gamma$ be $L$. Define $\lambda:[a, b] \rightarrow[0, L]$ by
$$\lambda(s)=L_C(a, s)=\int_a^s\left|\gamma^{\prime}(t)\right| d t$$
Then
$$\lambda^{\prime}(t)=\left|\gamma^{\prime}(t)\right| \neq 0$$
so $\lambda$ is a strictly increasing function on $[a, b]$ with a continuous derivative $\lambda^{\prime}$ on $[a, b]$, where $\lambda(a)=0, \lambda(b)=L$. It is regular since $\lambda^{\prime}(t) \neq 0$. It therefore has a strictly increasing inverse function $\rho=\lambda^{-1}$. Now $\rho:[0, L] \rightarrow[a, b]$ and has continuous derivative
$$\rho^{\prime}(t)=1 / \lambda^{\prime}(t) \neq 0$$
for $a<t<b$, so $\rho$ is also regular.
The path
$$\tau=\gamma \circ \rho:[0, L] \rightarrow \mathbb{C}$$
is regular, and
$$\tau(\lambda(t))=\gamma \circ \rho \circ \lambda(t)=\gamma(t)$$

## 数学代写|复分析代写Complex analysis代考|Regular Paths and Curves

6.20.定义让$\gamma:[a, b] \rightarrow \mathbb{C}$成为一条平坦的道路。

6.21.定义规则路径是平滑的路径$\gamma:[a, c] \rightarrow \mathbb{C}$，使得$\gamma^{\prime}(t) \neq 0$适用于所有$t \in[a, b]$。

2.4节中的标准路径$L(t)$(线)和$C(t)$(圆)是规则的。

$$\tau\left(t_0+h\right)=\gamma\left(t_0\right)+h \gamma^{\prime}\left(t_0\right) \quad(h \in \mathbb{R})$$

$$\gamma\left(t_0+h\right) \quad\left(t \text { near } t_0\right)$$

## 数学代写|复分析代写Complex analysis代考|Parametrisation by Arc Length

6.23.定义设$\gamma:[a, b] \rightarrow \mathbb{C}$为规则路径，图像曲线为$C$。让$t_0, t_1 \in[a, b]$用$t_0 \leq t_1$，让$\gamma\left(t_0\right)=c, \gamma\left(t_1\right)=d$。那么$C$中从$c$到$d$的弧长$L_C(c, d)$就是$\left.\gamma\right|{\left[t_0, t_1\right]}$的长度;也就是$$L_C(c, d)=\int{t_0}^{t_1}\left|\gamma^{\prime}(t)\right| \mathrm{d} t$$

$$\lambda(s)=L_C(a, s)=\int_a^s\left|\gamma^{\prime}(t)\right| d t$$

$$\lambda^{\prime}(t)=\left|\gamma^{\prime}(t)\right| \neq 0$$

$$\rho^{\prime}(t)=1 / \lambda^{\prime}(t) \neq 0$$

$$\tau=\gamma \circ \rho:[0, L] \rightarrow \mathbb{C}$$

$$\tau(\lambda(t))=\gamma \circ \rho \circ \lambda(t)=\gamma(t)$$

## MATLAB代写

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