TITLE: Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
QUESTION [46 upvotes]: I have spent some time using gp-pari. There is, of course, a formal power series solution to
$ f(f(x)) = \sin x.$  It is displayed below, identified by the symbol $g$ because I am not entirely sure whether it is a function of anything. 
On the other hand, should the coefficients continue to (by and large) decrease, this suggests a nonzero radius of convergence. If the radius of convergence is nonzero, then inside that, not only is a function defined and, you know, analytic, but the functional equation is satisfied. Indeed, all that is necessary is radius of convergence strictly larger than $\frac{\pi}{2}$ owing to certain symmetries. For instance, given my polynomial $g,$ it seems we have $g=1$ at about $x \approx 1.14.$ Then we seem to have a local maximum at $x =\frac{\pi}{2},$ and apparently there   $g \approx 1.14,$ strictly larger than 1 which is an important point.  So everything would fall into place with  large enough nonzero radius of convergence.    
$$
\begin{array}{lll}
g & = & x - \frac{x^3 }{ 12} -  \frac{x^5 }{ 160} -  \frac{53  x^7 }{ 40320}   -  \frac{23  x^9 }{71680} -  \frac{92713  x^{11}}{1277337600}  - \\\
 & & \\\
& & \frac{742031  x^{13} }{79705866240} +  \frac{594673187  x^{15} }{167382319104000} +  \frac{329366540401  x^{17} }{91055981592576000}  + \\\
& & \\\
& &     \frac{104491760828591  x^{19} }{62282291409321984000} +   \frac{1508486324285153   x^{21} }{4024394214140805120000} + \cdots
\end{array}
$$
Note that the polynomial $g$ is smaller than $x$ but larger that $\sin x,$ for, say,
 $0 < x \leq \frac{\pi}{2}.$ 
So, that is the question, does the formal power series beginning with $g$ converge anywhere
other than $x = 0$?
EDIT:  note that the terms after the initial $x$ itself have all turned out to be
$$  \frac{a_{2 k + 3} x^{2 k + 3}  }{2^k ( 2 k + 4)!} $$
where each $a_{2 k + 3}$ is an integer. This much seems provable, although I have not tried yet.
EDIT, Friday 12 November 2010. It now seems really unlikely that this particular problem gives an analytic answer. I suspect that the answer is $C^\infty$ and piecewise analytic, with failure of analyticity at only the points "parabolic" where the derivative has absolute value as large as 1, those points being $0,\pi, 2 \pi, \ldots.$ However, we need the anchor point at the fixpoint 0, otherwise how to begin? And I do think the power series will serve as an asymptotic expansion around 0.
Given the problem with the size of the derivative, now I am hoping for great things, and an obviously periodic and analytic solution, to the easier variant $f(f(x)) = g(x) = (1/2) \sin x.$ I would like both a nice power series and a nice answer by methods summing iterates  $ g^{[k]}(x),$ which for the moment is an entirely mysterious method to me, but attractive for periodic target functions as periodicity would be automatic.

REPLY [4 votes]: This is not a new answer but additional info for Will Jagy's answer about the computation of the Abel-function with the method of J. Écalle. 

[update] With some examples it comes out, that the "standard" computation of the fractional iteration via the formal power series
for the iterative logarithm  and its truncation to the leading terms
provides the same values as Écalle's method using the Abel function
as described below. With the same truncation to 64 terms of the power
series and the same shift of z_0 to z_h towards zero the
difference between the two methods is smaller than 1e-40 , which
is the accuracy which is also achieved by each method alone. [end
update]

I computed the formal Laurent-series for the Écalle-type Abel-function using Pari/GP to 509 coefficients in exact rational numbers which means the coefficients from $z^{-2}$ to $z^{506}$.            
The last coefficient has numerator with  1423 digits and denominator with 1247 digits amounting to an absolute value of about 175 digits, roughly -2.66945040282 E175 , so the series has in the same way as the comparable series for the fractional iteration of $\exp(x)-1$ zero-radius of convergence and when we plot the curve showing the number of digits of the nonzero coefficients by $\log_{10}(|a_k|)$ we get the typical shape of the upright hockeystick.         
Here are the leading 11 nonzero terms of the Laurent series for the Abel-function (which I call here "incomplete Abel function" so far because the "complete" Abel-function needs also the term for the logarithm and the term for the iteration-height h (this is index n in Will's answer):              
  Laurent-series in z:                             3  *z^-2
                                           + 79/1050  *z^2
                                           + 29/2625  *z^4
                                    + 91543/36382500  *z^6
                              + 18222899/28378350000  *z^8
                             + 88627739/573024375000  *z^10
                        + 3899439883/142468185234375  *z^12
              - 32544553328689/116721334798818750000  *z^14
                 - 4104258052789/1554729734250000000  *z^16
 - 119345896838809094501/141343700374629565312500000  *z^18
      + 745223773943295679/3505548124370772949218750  *z^20
             + O(z^22)

$ \qquad \qquad$ (remark: see an overview characterizing the growthrate at end (§2)) 
This gives the "incomplete Abel function" in terms of its coeffs up to some truncation n (all formulae in Pari/GP notation):
 abel_inc(z,n=64) = sum(k=1,n, coeff[k]*z^(k-3) )     

The complete Abel-function is then :                 
 {abel(z,h=32,n=64) = local(z_h,a); \\ give some sufficient default values 
                                    \\ in h and n for the required numerical 
                                    \\ precision of the approximate results
        z_h = sin_iter(z,h);        \\ sin_iter prev. defined as iterable sin()
      a = abel_inc(z_h,n) + 6/5*log(z_h) - h ;
      return(a); }                      

The inverse abel-function must be implemented by some root-solver. In Pari/GP I used the following, where the inverse Abel-function is included in the body of the full fractionally-iterable sin_h() function:
    {sin_h (h = 0,z_0=1) = local(a_0,z_h,a_h);  \\ restriction abs(h)<1 
               a_0 = abel(z_0, 32, 64);         \\ get the Abel-value for z_0
                                                \\ with meaningful precision
               a_h = a_0 + h ;                  \\ comp Abel-value for z_h
                                 \\ the following is the implementation of
                                 \\ the inverse Abel-function:
         z_h = solve(z = sin(z_0),z_0, abel(z,32,64) - a_h);
         return(z_h); }

The following is done to apply the above to some example, reproducing additivity of the iteration heights 0.5 and 0.5 to integral height 1 with more than 40 digits precision:          
                               \\  Pari-output
   z_0  = 1                    \\  %529 = 1
   z_05 = sin_h(0.5,z_0 )      \\  %530 = 0.908708429743
   z_1  = sin_h(0.5,z_05)      \\  %531 = 0.841470984808
   z_1 - sin(z_0)              \\  %532 = -6.38920219348 E-42

Below I show the recomputed list of computations in Will's answer with 40 digits correct:             
   step   z0=Pi/2 - step   abel(z0)   z05=sin_h(0.5,z0) z1=sin_h(0.5,z05)  z1 - sin(z0)
   0.00   1.57079632679  2.08962271973   1.14017947617  1.00000000000   -2.89445031739E-41
   0.05   1.52079632679  2.09536408453   1.13806963935  0.998750260395  -2.86591796888E-41
   0.10   1.47079632679  2.11273622895   1.13178674818  0.995004165278  -2.78164697945E-41
   0.15   1.42079632679  2.14218948912   1.12146458427  0.988771077936  -2.64553725829E-41
   0.20   1.37079632679  2.18449553252   1.10730765183  0.980066577841  -2.46383393292E-41
   0.25   1.32079632679  2.24078077607   1.08956885996  0.968912421711  -2.24476553049E-41
   0.30   1.27079632679  2.31257688904   1.06852649593  0.955336489126  -1.99807394218E-41
   0.35   1.22079632679  2.40189260763   1.04446448663  0.939372712847  -1.73446474837E-41
   0.40   1.17079632679  2.51131312355   1.01765794736  0.921060994003  -1.46500647333E-41
   0.45   1.12079632679  2.64413616528  0.988364216777  0.900447102353  -1.20050550750E-41
   0.50   1.07079632679  2.80455803137  0.956818478819  0.877582561890  -9.50882282773E-42
   0.55   1.02079632679  2.99792899241  0.923232674366  0.852524522060  -7.24576289372E-42
   0.60  0.970796326795  3.23110684637  0.887796468526  0.825335614910  -5.28014362671E-42
   0.65  0.920796326795  3.51295197372  0.850679308887  0.796083798549  -3.65188391373E-42
   0.70  0.870796326795  3.85503037983  0.812032915560  0.764842187284  -2.37402622132E-42
   0.75  0.820796326795  4.27262886030  0.771993802047  0.731688868874  -1.43260703471E-42
   0.80  0.770796326795  4.78624925852  0.730685613103  0.696706709347  -7.89576195851E-43
   0.85  0.720796326795  5.42385666222  0.688221187210  0.659983145885  -3.89074331205E-43
   0.90  0.670796326795  6.22434753781  0.644704322722  0.621609968271  -1.66626510284E-43
   0.95  0.620796326795  7.24305478745  0.600231264287  0.581683089464  -5.96979699941E-44
   1.00  0.570796326795  8.56077779381  0.554891942675  0.540302305868  -1.69831000319E-44

A picture of $y=\sin(x)$, the half-iterate $y=\sin^{\circ 0.5}(x)$ , $y=\sin^{\circ 1/3}(x)$ and $y=x$:                

Remark: at x, where sin(x)=0 the computation of the Abel-function runs in singularities and the value for the function is (interpolated from its neighbourhood) assumed to be zero.

In the Abel-function abel(z,h=32,n=64)=... there is the parameter h which allows to control the quality of approximation. The formal exact solution is given as limit when h goes to infinity - but we use only finite approximations here. They key is, that h controls the implicite iteration of the argument z towards the fixpoint zero, so the numerical evaluation of the (truncated to n coefficients) Laurent-series gives a better approximation to the true value - although actually the convergence-radius is still zero! The purpose of those iterations shifting z_h towards zero is to shift the position, from where the Laurent series with the argument z_h begins to diverge, to higher indexes and thus to get more accuracy. A combination of h=32 and n=64 for arguments $|z| \le 1$ is apparently enough for 40 correct digits. (see remark (§1))
Finally, to show the effect of the h=32 iteration at work I provide below the partial sums of the Laurent-series for z=1 in comparision to h=4.      
In the first example I use h=4 and in the second example I use h=32 .
In the table k is the index of the coefficient up to where the partial sums are computed. ps_k indicates the partial sum using z_h which is the h 'th iterate from z_0=1 . But for convenience the term for the logarithm and the h-term are always included so we can compare the sum up to this term with the accurate value a_1 for the Abel-function at z_1 :                 
   k    ps_k              error:  a_1 - ps_k    iteration height h=4
   0  3.05810608515        -0.0315166345810
   2  3.05810608515        -0.0315166345810
   4  3.08773833843       -0.00188438129901
   6  3.08945198975      -0.000170729978211
   8  3.08960570369     -0.0000170160371392
  10  3.08962115403    -0.00000156570332243
  12  3.08962261968   -0.000000100050871450
  14  3.08962272183  0.00000000210083986271
  16  3.08962272142  0.00000000169099804938
  18  3.08962271989       1.62746538183E-10
  20  3.08962271970      -2.97721970306E-11
  ... 
  50  3.08962271973      -3.98604755990E-18
  52  3.08962271973       7.74229820435E-19
  54  3.08962271973       1.21098784690E-18
  56  3.08962271973      -6.22150631919E-20
  58  3.08962271973      -3.98357488277E-19
  60  3.08962271973      -3.38541477910E-20
  62  3.08962271973       1.42850133024E-19

We see, that with iteration-height h=4 we arrive at an absolute error smaller than 1e-18 at the 64. term.
And below, iteration-height h=32 provides accuracy to an absolute error of smaller than 1e-40 with that 64 terms used:
   k    ps_k              error:  a_1 - ps_k    iteration height h=32
   0  3.08337725463         -0.00624546510435
   2  3.08337725463         -0.00624546510435
   4  3.08954701281       -0.0000757069234782
   6  3.08962130264      -0.00000141708899538
   8  3.08962269011    -0.0000000296188288642
  10  3.08962271915  -0.000000000581829933894
  12  3.08962271972        -8.31025344698E-12
  ... ...                  ...
  52  3.08962271973        -3.06907747463E-37
  54  3.08962271973         5.27409063179E-37
  56  3.08962271973         2.10119895640E-38
  58  3.08962271973        -6.82487772781E-39
  60  3.08962271973        -5.39925105785E-40
  62  3.08962271973         9.44571568505E-41


(§1): a Noerlund-summation, as I have it proposed in some treatizes for the evaluation of the fractional iterates of the $\exp(x)-1$ might give arbitrary approximations as well, but it seems now to me that such a summation-procedure were at most needed here for theoretical reasons to prove the summability of the Laurent-series for the Abel-function.
(§2): A short overview over the first 512 coefficients of the abel_inc()-series:
 index      value             index   value            index    value               index    value
    0           3.000000000   47                     0   92       -0.005185699555  496       4.633504372E168 
    1                     0   48  -0.00000003870320993   93                     0  497                     0
    2                     0   49                     0   94         0.01347223160  498      -4.983759375E169
    3                     0   50  0.000000006386371562   95                     0  499                     0
    4         0.07523809524   51                     0   96         0.03559427183  500      -8.187596780E170
    5                     0   52   0.00000006229599636   97                     0  501                     0
    6         0.01104761905   53                     0   98        -0.06747379661  502       8.333103850E171
    7                     0   54   0.00000001451248843   99                     0  503                     0
    8        0.002516127259   55                     0  100         -0.2528544049  504       1.467790435E173
    9                     0   56   -0.0000001074166810  101                     0  505                     0
   10       0.0006421408926   57                     0  102          0.3439480705  506      -1.412786474E174
   11                     0   58  -0.00000007200630916  103                     0  507                     0
   12       0.0001546666126   59                     0  104           1.879638019  508      -2.669450403E175
   13                     0   60    0.0000001982539503  105                     0  ...          ...
   14      0.00002737060121   61                     0  106          -1.706858981
   15                     0   62    0.0000002440284845  107                     0
   16   -0.0000002788226624   63                     0  108          -14.69827943
   17                     0   64   -0.0000003845753696  109                     0
   18    -0.000002639853064   65                     0  110           7.295584305
   19                     0   66   -0.0000007917263057  ...             ...         
   20   -0.0000008443665796   67                     0
   ...                  ...   ...                  ...