Summary

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• Compute the coefficients of the Jungreis function

• 
  Uniformisation of the interior of Mandelbrot set components using multiplier map: internal rays , internal coordinate

c src code

// https://gist.github.com/Gro-Tsen/9f690a9461c99abb4a01249b1838b548
// Gro-Tsen http://www.madore.org/~david/
// Compute the coefficients of the Jungreis function, i.e., the
// Fourier coefficients of the harmonic parametrization of the
// boundary of the Mandelbrot set, using the formulae given in
// following paper: John H. Ewing & Glenn Schober, "The area of the
// Mandelbrot set", Numer. Math. 61 (1992) 59-72 (esp. formulae (7)
// and (9)).  (Note that their numerical values in table 1 give the
// coefficients of the inverse series.)

// The coefficients betatab[m+1][0] are the b_m such that
// z + sum(b_m*z^-m) defines a biholomorphic bijection from the
// complement of the unit disk to the complement of the Mandelbrot
// set.  The first few values are:
//   {-1/2, 1/8, -1/4, 15/128, 0, -47/1024, -1/16, 987/32768, 0, -3673/262144}

// Formula: b_m = beta(0,m+1) where beta(0,0)=1,
// and by downwards induction on n we have:
// beta(n-1,m) = (beta(n,m) - sum(beta(n-1,j)*beta(n-1,m-j), j=2^n-1..m-2^n+1) - beta(0,m-2^n+1))/2 if m>=2^n-1, 0 otherwise
// (beta(n,m) is written betatab[m][n] in this C program).

#include <stdio.h>
#include <math.h>
#include <float.h>
#include <assert.h>

#ifndef NBCOEFS
#define NBCOEFS 8193
#endif

#ifndef SIZEN
#define SIZEN 14
#endif

#if ((NBCOEFS+4)>>SIZEN) != 0
#error Please increase SIZEN
#endif

double betatab[NBCOEFS+1][SIZEN];

int
main (void)
{
  for ( int n=0 ; n<SIZEN ; n++ )
    betatab[0][n] = 1;
  for ( int m=1 ; m<=NBCOEFS ; m++ )
    {
      for ( int n=SIZEN ; n>=1 ; n-- )
	{
	  if ( m < (1<<n)-1 )
	    betatab[m][n-1] = 0;
	  else
	    {
	      double v;
	      assert (n < SIZEN);
	      v = betatab[m][n];
	      for ( int k=((1<<n)-1) ; k<=m-(1<<n)+1 ; k++ )
		v -= betatab[k][n-1] * betatab[m-k][n-1];
	      v -= betatab[m-(1<<n)+1][0];
	      betatab[m][n-1] = v/2;
	    }
	}
      printf ("%d\t%.17e\t%a\n", m-1, betatab[m][0], betatab[m][0]);
    }
}

Maxima source code

 /*  batch file for maxima 
 uses :
 - symmetry around horizontal ( 0X ) axis
 - Psi_M function to map conjugate plane to parameter plane
 - jungreis algorithm to 
 time :3818
 */ 

 start:elapsed_run_time ();
 jMax:50; /* precision  = proportional to details and time of computations  */
 iMax:200; /* number of points to draw */
 iMaxBig:400;

 /* computes b coefficient of Jungreis function*/
 betaF[n,m]:=block
 (
 [nnn:2^(n+1)-1],
 if m=0
 then 1.0
 else if ((n>0) and (m < nnn)) 
  then 0.0
  else (betaF[n+1,m]- sum(betaF[n,k]*betaF[n,m-k],k,nnn,m-nnn)-betaF[0,m-nnn])/2.0
 )$
 b[m]:=betaF[0,m+1]$

 /* -------------------------------*/
 /* Power of w to j */
 wn[w,j]:= if j=0 then 1 else w*wn[w,j-1]$

 /* ---------Jungreis function ;  c = Psi_M(w) ----------------------------- */
 Psi_M(w):=w + sum(b[j]/wn[w,j],j,0,jMax)$

 /* exponential for of complex number with angle in turns */
 GiveCirclePoint(t):=R*%e^(%i*t*2*%pi)$ /* gives point of unit circle for angle t in turns */

 GiveWRayPoint(R):=R*%e^(%i*tRay*2*%pi)$ /* gives point of external ray for radius R and angle tRay in turns */

 NmbrOfCurves:9;

 /* coordinate of w-plane  not c-plane */
 R_max:1.5;
 R_min:1;
 dR:(R_max-R_min)/NmbrOfCurves; /* for circles */
 dRR:(R_max-R_min)/iMax; /* for rays */ 

 /* --------------------------------------f_0 plane = w-plane -----------------------------------------*/

 /*-------------- unit circle ------------*/
 R:1;
 circle_angles:makelist(i/iMax,i,0,iMax/2)$
 CirclePoints:map(GiveCirclePoint,circle_angles)$ 

 /* ---------------external circles ---------*/
 circle_radii: makelist(R_min+i*dR,i,1,NmbrOfCurves)$ 
 circle_angles:makelist(i/iMaxBig,i,0,iMaxBig/2)$
 WCirclesPoints:[]$
 for R in circle_radii do 
 WCirclesPoints:append(WCirclesPoints,map(GiveCirclePoint,circle_angles))$

 /* -------------- external w rays -------------*/
 ray_radii:makelist(R_min+dRR*i,i,1,iMax);
 ray_angles:[0,1/3,1/7 , 2/7 ,3/7  ]; /* list of angles < 1/2 of root points */
 WRaysPoints:[];
 for tRay in ray_angles do 
 WRaysPoints:append(WRaysPoints,map(GiveWRayPoint,ray_radii));

 /* -------------------------parameter plane = c plane -----------------------------------*/
 MPoints:map(Psi_M,CirclePoints); /* Mandelbrot set points */

 CRaysPoints:map(Psi_M,WRaysPoints); /* external z rays */

 Equipotentials:map(Psi_M,WCirclesPoints);

 /* add points below horizontal axis */
 for w in CirclePoints do CirclePoints:cons(conjugate(w),CirclePoints);
 for w in WRaysPoints do WRaysPoints:cons(conjugate(w),WRaysPoints);
 for w in WCirclesPoints do WCirclesPoints:cons(conjugate(w),WCirclesPoints);
 for c in MPoints do MPoints:cons(conjugate(c),MPoints);
 for c in CRaysPoints do CRaysPoints:cons(conjugate(c),CRaysPoints);
 for c in Equipotentials do Equipotentials:cons(conjugate(c),Equipotentials);

 /* time */
 stop:elapsed_run_time ();
 time:fix(stop-start); 

 /* ---------------- draw *--------------------------------------------------------------------------*/

 load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
 draw(file_name = "jung50es_2",
  pic_width=1000, 
  pic_height= 500,
  terminal  = 'png,
  columns  = 2,
  gr2d(title = " unit circle with external rays & circles ",
  point_type = filled_circle,
  points_joined =true,
  point_size    =  0.34,
  color         = red,
  points(map(realpart, CirclePoints),map(imagpart, CirclePoints)),
  points_joined =false,
  color         = green,
  points(map(realpart, WCirclesPoints),map(imagpart, WCirclesPoints)),
  color         = black,
 points(map(realpart, WRaysPoints),map(imagpart, WRaysPoints))
 ),
 gr2d(title      = "Parameter plane : Image under Psi_M(w) ",
 points_joined =true,
 point_type = filled_circle,
 point_size    =0.34,
 color         = blue,
 points(map(realpart, MPoints),map(imagpart, MPoints)),
 points_joined =false,
 color         = green,
 points(map(realpart, Equipotentials),map(imagpart, Equipotentials)),
 color         = black,
 points(map(realpart, CRaysPoints),map(imagpart, CRaysPoints))
 ) 
 );

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