# 数学代写|复分析作业代写Complex function代考|Further Properties of Lengths

#### Doug I. Jones

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## 数学代写|复分析作业代写Complex function代考|Further Properties of Lengths

Recall from Definition 2.27 that paths can be added together if their start and end points match up correctly. Using approximating polygons, it is easy to prove that the length function $L$ is additive:
PROPOSITION 6.16. If $\gamma=\gamma_1+\cdots+\gamma_n$ and $L\left(\gamma_r\right)$ exists for $1 \leq r \leq n$, then
$$L(\gamma)=L\left(\gamma_1\right)+\cdots+L\left(\gamma_n\right)$$
It is important to understand that the length of a path need not be the same as the length of its image curve. In particular, this happens whenever the path ‘doubles back on itself’, as in the following example.

Next, we consider how a change of parameter affects the length of a smooth path.
DEFINITION 6.18. Smooth paths $\gamma:[a, b] \rightarrow \mathbb{C}, \lambda:[c, d] \rightarrow \mathbb{C}$ with the same image curve $C$ are smoothly equivalent parametrisations of $C$ if there is a smooth function $\rho:[a, b] \rightarrow[c, d]$ with non-zero derivative $\rho^{\prime}(t) \neq 0$ for $t \in[a, b]$, where $\rho(a)=c$, $\rho(b)=d$, and $\gamma=\lambda \circ \rho$.

This relationship is an equivalence relation, because $\rho^{-1}$ exists, is smooth, and also has non-zero derivative, by the inverse function theorem. A smoothly equivalent parametrisation can be imagined as tracing the same curve in the same direction but at different speeds.

PROPOSITION 6.19. If two smooth paths $\gamma:[a, b] \rightarrow \mathbb{C}, \lambda:[c, d] \rightarrow \mathbb{C}$ are smoothly equivalent parametrisations of the same curve $C$, then they have the same length: $L(\gamma)=L(\lambda)$.

Proof. We have $\gamma=\lambda \circ \rho$ where $\rho:[a, b] \rightarrow[c, d]$ is a strictly increasing real function with continuous derivative $\rho^{\prime}$ on $[a, b]$. Therefore $\rho^{\prime}(t) \geq 0$, so $\rho^{\prime}(t)=\left|\rho^{\prime}(t)\right|$. Let $s=\rho(t)$. Then as $t$ increases from $a$ to $b, s$ increases from $c$ to $d$ and we can substitute $\mathrm{d} s=\rho^{\prime}(t) \mathrm{d} t$ in the integral. The length of the path $\gamma$ is therefore
\begin{aligned} L(\gamma) & =\int_a^b\left|\gamma^{\prime}(t)\right| \mathrm{d} t \ & =\int_a^b\left|(\lambda \circ \rho)^{\prime}(t)\right| \mathrm{d} t \ & =\int_a^b \mid\left(\lambda^{\prime}(\rho(t))|| \rho^{\prime}(t) \mid \mathrm{d} t\right. \ & =\int_a^b \mid\left(\lambda^{\prime}(\rho(t)) \mid \rho^{\prime}(t) \mathrm{d} t\right. \ & =\int_a^b\left|\lambda^{\prime}(s)\right| \mathrm{d} s=L(\lambda) \end{aligned}

## 数学代写|复分析作业代写Complex function代考|Lengths of More General Paths

You may be wondering why we require $\gamma$ to be smooth in the definition of length in Definition 6.15? Proposition 6.16 applies to contours made from several smooth pieces, which need not fit together smoothly where they join, so paths that are not smooth can have meaningful lengths. Obviously, Definition 6.15 cannot be used for continuous paths, because it involves the derivative of $\gamma$. But the definition ‘supremum of lengths of all approximating polygons’, which occurs in the alternative treatment of Section 6.3, makes sense for any path, without assuming smoothness. However, this definition has its own awkward features. Sometimes it goes wrong in the mild sense that the entire path has infinite length, as in Examples 6.9 and 6.10. However, it gets much worse. Using the ‘approximating polygon’ definition for length, there are continuous paths $\gamma:[a, b] \rightarrow \mathbb{C}$ with the disturbing property that the length of any segment of the path, between points $c<d$ in $[a, b]$, is infinite.

In fact, we have already met just such a path: the graph of the blancmange function, Figure 4.3. Using dyadic rationals, it is straightforward to construct a sequence of partitions of $[c, d]$ such that the corresponding polygons have lengths tending to infinity. In fact, it is enough to do this for the graph of the blancmange function on the interval $[0,1]$, because we have already pointed out that this graph contains arbitrarily small copies of the graph of the blancmange function, and a small number times infinity is still infinity. These copies are usually distorted by an affine transformation, but such a distortion keeps the length infinite. Similar remarks apply to space-filling curves, and to standard fractals such as the snowflake curve, Mandelbrot [13], and Falconer [4].

In other words, a path that is continuous but not smooth may not have a meaningful length. This is probably counterintuitive, because we have been trained from an early age to assume that every linear object does have a length. The ancient Greeks worried about the length of the circumference of a circle without ever defining what they were worrying about. It seemed obvious to them that any arc of a circle must have a length; their main aim was to find out what that length is. To do so, they tacitly assumed several plausible properties, such as finding the length by using polygons to approximate the circumference. Eventually mathematicians tightened up the logic, leading to Definition 6.8 above, and understood what can go wrong.

To complete the story, smoothness is not actually necessary for a path or curve to have a well-defined length. There is a more general notion of a rectifiable curve, for which the ‘approximating polygon’ definition is entirely satisfactory. However, the smooth case is all we need in this book.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Further Properties of Lengths

$$L(\gamma)=L\left(\gamma_1\right)+\cdots+L\left(\gamma_n\right)$$

6.18.定义平滑的路径 $\gamma:[a, b] \rightarrow \mathbb{C}, \lambda:[c, d] \rightarrow \mathbb{C}$ 用相同的图像曲线 $C$ 平滑等价的参数化是 $C$ 如果有一个平滑函数 $\rho:[a, b] \rightarrow[c, d]$ 非零导数 $\rho^{\prime}(t) \neq 0$ 为了 $t \in[a, b]$，其中 $\rho(a)=c$， $\rho(b)=d$，和 $\gamma=\lambda \circ \rho$．

\begin{aligned} L(\gamma) & =\int_a^b\left|\gamma^{\prime}(t)\right| \mathrm{d} t \ & =\int_a^b\left|(\lambda \circ \rho)^{\prime}(t)\right| \mathrm{d} t \ & =\int_a^b \mid\left(\lambda^{\prime}(\rho(t))|| \rho^{\prime}(t) \mid \mathrm{d} t\right. \ & =\int_a^b \mid\left(\lambda^{\prime}(\rho(t)) \mid \rho^{\prime}(t) \mathrm{d} t\right. \ & =\int_a^b\left|\lambda^{\prime}(s)\right| \mathrm{d} s=L(\lambda) \end{aligned}

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