|exp(z)-1|在|z|=1的最值?

hbghlyj

$t\in[0,2\pi)$, 求证\[\sqrt{\Bigl(\exp (\cos (t)) \cos (\sin (t))-1\Bigr)^2+\Bigl(\exp (\cos (t))\sin (\sin (t))\Bigr)^2}\in[1-e^{-1},e-1]\]
问题来自url=forum.php?mod=viewthread&tid=9848

参数曲线$$\cases{x=\exp (\cos (t)) \cos (\sin (t))-1\\y=\exp (\cos (t))\sin (\sin (t))}\quad t\in[0,\pi]$$

hbghlyj

验证

1. Maximize[(Exp[Cos[t]]Cos[Sin[t]]-1)^2+(Exp[Cos[t]]Sin[Sin[t]])^2,t]

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$1 - 2e + e^2,\;t \to 0$