Exercise 6.18

Since for any $x\in R$ we have $|e^{\mathit{ix}}|=|\cos <!--\; FUNCTION\; APPLICATION\; -->x+i\sin <!--\; FUNCTION\; APPLICATION\; -->x|={\cos <!--\; FUNCTION\; APPLICATION\; -->}^{2}x+{\sin <!--\; FUNCTION\; APPLICATION\; -->}^{2}x=1$, the images of all three curves lie on the unit circle $S^1$. And if $z=\cos <!--\; FUNCTION\; APPLICATION\; -->x+i\sin <!--\; FUNCTION\; APPLICATION\; -->x$ is an arbitrary point of $S^1$, $0\le x<2\pi$, then $z={\gamma }_1(x)={\gamma }_2(x∕2)$, so the images of ${\gamma }_1$ and ${\gamma }_2$ are all of $S^1$. Also, since

by the intermediate-value theorem (Theorem 4.23) if $-1.33\le y\le 6.18$, there is a $0\le t\le 2\pi$ such that $2\mathit{\pi t}\sin <!--\; FUNCTION\; APPLICATION\; -->(1∕t)=y$, so that ${\gamma }_3(t)=e^{\mathit{iy}}=z$. Hence the image of ${\gamma }_3$ is also all of $S^1$.

Since $|{\gamma }_1(t)|=|i{e}^{\mathit{it}}|=1$ and $|{\gamma }_2(t)|=|2i{e}^{2\mathit{it}}|=2$, by Theorem 6.27 we have

For ${\gamma }_3$ we have

Let $a_n=(2n+1)\pi$, $b_n=(2n+1∕2)\pi$ for any integer $n\ge 1$. Then $[a_n^{-1},b_n^{-1}]$ is a subinterval of $[0,2\pi ]$ on which $\sin <!--\; FUNCTION\; APPLICATION\; -->(1∕t)\ge 0$ and $\cos <!--\; FUNCTION\; APPLICATION\; -->(1∕t)\le 0$. Hence the length of ${\gamma }_3$ on this subinterval is

Since $\Lambda ({\gamma }_3)$ must be larger than the divergent sum of such terms, ${\gamma }_3$ is not rectifiable.