We define the iterated sine function as : $$ \sin^n(x) = \sin(\sin(.... \sin(x)))\:\:n\:\text{times.} $$

We know the "Frequency Modulation" formula based on Bessel functions :$$ \sin( p\, \sin(x) ) = \sum_{k=1, \,odd}^{\infty} 2 \, J_k(p) \, \sin( k \,x) \qquad, J_k(p) : \text{Bessel function of the first kind}$$

if we write the different harmonics of $\sin^n$ as an (infinite) vector over the $\sin(x), \sin(3x), \sin(5x),...$ base, we have:

$$ [\sin^n] = 2^{(n-1)} \begin{bmatrix} J_1(1) & J_1( 3) & \cdots & J_1(col)& \cdots \\ J_3(1) & J_3( 3) & \cdots & J_3(col)& \cdots\\ \vdots & \vdots & \vdots & \ddots\\ J_{line}(1) & J_{line}( 3) & \cdots &J_{line}(col)& \cdots\\ \vdots& \vdots& \vdots& \vdots& \ddots \end{bmatrix}^{(n-1)} . \begin{bmatrix} 1 \\0 \\ 0 \\ \vdots\end{bmatrix}$$ which gives already excellent approximations of $ \sin^n$: $$ \sin^{(2)}(x) = 0.8773 \sin(x) + 0.0411 \sin(3x) +0.0005 \sin(5x)+...$$ $$ \sin^{(^1/_2)}(x) = 1.0945 \sin(x) - 0.00392 \sin(3x) + 0.0056 \sin(5x)+...$$ and shows that most of the energy stays in the first 2 harmonics. It even "finds": $$ \sin^0(x) = \sum_{k=1, \,odd}^{\infty} \frac{4}{k^2 \pi} \sin( k \pi / 2) \sin(kx) \qquad \text{(sawtooth wave)} $$ Questions:

  1. How to stabilize the diagonalization of this "Bessel Matrix" : practically it works well, but eigenvalues / vectors vary wildly when dimensions change. Is there a way to "order" the eigen values so the diagonalization process somehow converges ?
  2. Do I have a way to get an approximation of the coefficient of $\sin(x)$(resp $\sin(3x) $ etc. as a function of n ? Numerical approximation gives $\sin^n(x) = [0.67 \cdot 0.98^n + 0.34 \cdot 0.757^n ] \sin(x) + [0.12 \cdot 0.98^n - 0.10 \cdot 0.757^n - 0.17 \cdot 0.282^n ] \sin(3 x)+...$ but this seems very dependent upon the dimension used during diagonalization.
  3. How can I choose the dimension to "cap" the error on the iterate sin to a certain level, etc.

Anyway for me the best way to compute the iterated sin with only a few terms ... converges very fast over $\mathbb{R}$ and applicable also to iterated cos !

  • $\begingroup$ Just tried dimensions 4x4, 8x8, 16x16, 32x32 - looks very stable and promising. $\endgroup$ Oct 30, 2022 at 11:19
  • $\begingroup$ Pinged this question to the tetration-forum, Perhaps some expert from there enters discussion. See math.eretrandre.org/tetrationforum/… for (possible) discussion. $\endgroup$ Oct 30, 2022 at 19:25
  • $\begingroup$ @ your last sentence: how is this transferred to the iterated $\cos()$-function? $\endgroup$ Oct 30, 2022 at 19:26

1 Answer 1


About question 1), stabilizing the eigenvalues of diagonalization for changing dimensions.

Just to make this better visible, I show two pictures, where the sets of eigenvalues for dimensions $N \in \{4,8,16,32,64,128\}$ are shown.

The indexes are rescaled to the interval $0..1$. The modification of the sets of eigenvalues seems nicely tamed instead of chaotic to me:


Even more tamed, if the x-axis is logarithmically scaled:


Perhaps from the second picture one can formulate a guess-function.


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