TITLE: About $\sum \sin(\kappa \sin (\kappa \sin(\cdots \sin \kappa x)\cdots)$
QUESTION [6 upvotes]: I have proved that, given $|\kappa|<1$, the function:
$$f(x)=\sum_{n\geqslant 1} \underbrace{\sin(\kappa \sin (\kappa \sin (\cdots \sin}_n\, \kappa x)\cdots)$$
Is defined for all real $x$ (the series converges everywhere). My question is, what does the function look like? I don't have the software to plot it, and I can't figure it out on paper. Sorry if the question is a little vague.

REPLY [3 votes]: The question asks for what the function looks like.  Sadly the answer is rather dull.  I used Geogebra to graph the first seven terms in the series and their sum for values of $\kappa$ that are controlled by a slider.  (Adding more terms does not seem to significantly change the appearance of the graph.)  That is, I plotted
$$f_1(x)=\sin(\kappa x)$$
$$f_2(x)=\sin(\kappa f_1(x))$$
$$f_3(x)=\sin(\kappa f_2(x))$$
$$\vdots$$
$$f_7(x)=\sin(\kappa f_6(x))$$
$$g(x)=f_1(x)+f_2(x)+\cdots +f_7(x)$$
The graph is shown below.  The blue curve is the sum; the black curves are the individual terms of the series.