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