A paper stated $| e^{iwx} - 1 | \le 2 |w| \, |x|$ without any further explanation. I struggle to find an explanation because Euler's formula doesn't help.
Can you give me a give why this statement might be true?
2026-03-25 15:57:31.1774454251
Explain why $| e^{iwx} - 1 | \le 2 |w| \, |x|$
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As Did has pointed, $wx\in\Bbb R$ is required (If $x > 0$ and $w = -i$, $|e^x - 1| = |e^{-i^2x} - 1| \le 2|i||x| = 2x$ is obviously false for $x$ big).
Wlog, we can suppose $w = 1$ and then $$ |e^{-ix} - 1| = |\cos x - i\sin x - 1| = {\sqrt{2 - 2\cos x}} = {\sqrt{4\sin^2(x/2)}}\le |x|. $$