Let

$$\begin{aligned} \varphi(w) = w + \frac{1}{w} \end{aligned}$$

Why is it true that

$$\begin{aligned} \varphi(w) : \{w : \lvert w \rvert > 1\} \rightarrow \mathbb{C} \,\backslash\, [-2,2] \end{aligned}$$

?

If $\lvert w \rvert \ge 2$, $\varphi(w)$ obviously excludes the interval $[-2,2]$.

If $w = 1$, $\varphi(w) = 2$.

Or more generally, if $\lvert w \rvert = 1$ or $w = e^{i\theta}$,

$$\begin{aligned} \varphi(w) &= e^{i\theta} + e^{-i\theta} \\ &= \cos{\theta} + i\sin{\theta} + \cos{\theta} - i\sin{\theta} \\ &= 2\cos{\theta} \\ \end{aligned}$$

$\therefore \,\,0 \le \lvert \varphi(w) \rvert \le 2$ when $\lvert w \rvert = 1$.

Observe that $\displaystyle \left| \varphi(w) \right|$ is monotonic increasing, which means the interval $[-2,2]$ is clearly excluded from $\displaystyle \varphi(w)$ as $\left| w \right| > 1$.

More info at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 9:34 PM