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My question is about composing sines. When you compose $\sin x$ ($x$ in radians of course) $k$ times, what is the maximum in terms of $k$?

For example, composing $2$ sines, as in $\sin(\sin x)$, attains a maximum of approximately $0.84147$ because $\sin(\sin\pi/2) = \sin1 \approx 0.84147$ and the value at $ x = \pi/2$ will obviously always be the maximum for any number of iterations.

So, the question is, does there exist an elementary function $f$ that takes the number of iterations and outputs the maximum of that many nested sines? That is, can $f$ given by $$ f(k)=\underbrace{\left(\sin\circ\cdots\circ\sin\right)}_{k \text{ compositions}}\left(\frac{\pi}{2}\right) $$ be expressed using elementary functions, excluding trigonometric functions?

I have fiddled around some but I cannot get the correct function to match a locus I made.

Brian61354270
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TigerGold
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1 Answers1

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Using Frequency Modulation and Bessel functions you can write $\sin^n$ as a matrix product ( see here ). It involves just sine functions with relatively elementary coefficients based on Besssel function you can find even in Excel !

And the maximum comes as powers of the eigenvalues of the Bessel matrix: $$ \sin^n(\pi/2) \approx \pi/2 - 0.2385 \ln( 10 n +1 ) \qquad \text{ for 0<n<1} $$ $$ \sin^n(\pi/2) \approx 1- 0.222 \ln( n ) \qquad \text{ for 1<n } $$

al4085
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