Find the fixed point of $\cos(x)$ (equivalently of $\cos(\cos(x))$) restricted to $[0,\frac \pi 2]$.

Asked Feb 14 '15 at 07:28

Active Feb 14 '15 at 08:45

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I've proved that $\cos(\cos(x))$ restricted to $[0,\frac \pi 2]$ is a contraction, which imply by Banach's fixed point theorem that it has a unique fixed point on this interval.

I've also proved that $\cos(x)$ must also have a unique fixed point on this interval, and that it must be the same as the one for $\cos(\cos(x))$.

But, how can I find this fixed point explicitly ?

asked Feb 14 '15 at 07:28

Shuzheng

5,821

Remember the proof of the Banach fixed point theorem? – Henno Brandsma Feb 14 '15 at 07:29
you might find this question of peripheral interest in connection with the cosine fixed point http://math.stackexchange.com/questions/622088/conjecture-a-supremum-property-of-the-cosine-fixed-point?rq=1 – David Holden Feb 14 '15 at 07:39
Well, there's the method of the proof of the theorem...pick any starting point in your interval, and iterate the function on that point. That sequence will converge to your fixed point – Alan Feb 14 '15 at 07:47
@Alan what happens if you try that? Try pressing cos repeatedly on a calculator, e.g. – Henno Brandsma Feb 14 '15 at 08:02
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Are you referring to the Dottie number, whose decimal expansion can be found here ? – Lucian Feb 14 '15 at 08:38
This might be of interest... – AvZ Feb 14 '15 at 08:48
So this number whatever it is, is not a "easy to find" number ? It might not be rational ? – Shuzheng Feb 14 '15 at 09:25
@user111854: as explained in the linked Wikipedia entry, it is a trascendental number as a consequence of the Lindemann-Weierstrass theorem. – Jack D'Aurizio Feb 14 '15 at 09:37

Find the fixed point of $\cos(x)$ (equivalently of $\cos(\cos(x))$) restricted to $[0,\frac \pi 2]$.

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