File:Conformal mapping from right half plane to unit circle.svg

• 
  Conformally map of upper half-plane to unit disk using ${\displaystyle z\mapsto {\frac {z-i}{z+i}}}$
• The point I is variable on [Oy) and (Γ) is a circle going through B and whose center is I. The picture is the inverse of the half-plane y<0 with respect to the circle (Γ)

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Map from z-plane ( without z=-1 ) to w-plane :

${\displaystyle h(z)={\frac {z-1}{z+1}},\;z\in \mathbb {C} \setminus \{-1\}}$

maps right half plane :

${\displaystyle \{z\in \mathbb {C} :Re(z)\geq 0\}}$

to unit disk about the origin :

${\displaystyle {\bar {D}}_{1}=\{z:|z|\leq 1\}.\,}$

/*

 b batch file for maxima



There are 2 complex planes :
* w-plane 
* z-plane 


z is a pont of z-planes
zz is a list of z points
zzz is a list of zz lists

1. draw cufves on z plane 
2. draw images of z-curves  under h  on w plane               
3. draw images of w-curves under hi on z plane 


http://math.stackexchange.com/questions/114733/mapping-half-plane-to-unit-disk

Adam Majewski
 
*/

kill(all);
remvalue(all);


/* definitions of  functions  */


/*  map from z to w : w=h(z) */
h(z):=if z=infinity then 1 else  (z-1)/(z+1); /* map from z plane to w plane : w = h(z) */

/*  inverse of h function   maps from w to z :  z:hi(w)  */ 
hi(w):= if w=1 then %i*infinity else rectform((1+w)/(1-w));

/* converts complex number into list for draw package  */
draw_format(z):= [float(realpart(z)),float(imagpart(z))];



compile(all);



print (" ============== compute ===============  ")$

/* curve is a list of points joned by lines */

 /* vertical z-lines */
 zz  : makelist (k*%i/10, k, -200, 200        )$ /* line re(z)=0  */
 zz1 : makelist (k*%i/10+1, k, -200, 200        )$ /* line re(z)=1 */
 zz2 : makelist (k*%i/10+2, k, -200, 200        )$ /* line re(z)=2  */

 /* horizontal z-lines */
 zzx  : makelist (k/10, k, 0, 400        )$ /* line im(z)=0 */
 zzx1 : makelist (%i+k/10, k, 0, 400        )$ /* line im(z)= 1   */
 zzx2 : makelist (2*%i+k/10, k, 0, 400        )$ /* line im(z) = 2  */
 zzxm1 : makelist (-%i+k/10, k, 0, 400        )$ /* line im(z)=-1  */
 zzxm2 : makelist (-2*%i+k/10, k, 0, 400        )$ /* line im(z)=-2  */
 
 /* --------- map from z to w plane using h  -------------------------------*/

 /* w-curves = images of verticla z-lines */
 ww  : map(h,zz )$ 
 ww1 : map(h,zz1)$
 ww2 : map(h,zz2)$
 /* w-curves = images of horizontal z-lines */
 wwx:map(h,zzx)$
 wwx1:map(h,zzx1)$
 wwx2:map(h,zzx2)$
 wwxm1:map(h,zzxm1)$
 wwxm2:map(h,zzxm2)$
  

 /*  map from w to z plane using hi */
 zzhi  : map(hi,ww)$
 zz1hi : map(hi,ww1)$
 zz2hi : map(hi,ww2)$

 zzxhi   : map(hi, wwx)$
 zzx1hi  : map(hi, wwx1)$
 zzx2hi  : map(hi, wwx2)$
 zzxm1hi : map(hi, wwxm1)$
 zzxm2hi : map(hi, wwxm2)$
 
 /* single important points */
 z0 : 0; /* origin z=0 */
 w0 : h(z0);
 z0hi : hi(w0);
 
 zi:infinity; /* point at infinity */
 wi:h(zi);
  

 

print ("--------------  convert lists of complex points to draw format lists ----------- ")$

 zz:map(draw_format,zz)$ 
 ww:map(draw_format,ww)$ 
 zzhi:map(draw_format,zzhi)$

 zz1:map(draw_format,zz1)$ 
 ww1:map(draw_format,ww1)$ 
 zz1hi:map(draw_format,zz1hi)$

 zz2:map(draw_format,zz2)$ 
 ww2:map(draw_format,ww2)$
 zz2hi:map(draw_format,zz2hi)$

 /* horizontal z-lines */
 zzx  : map(draw_format,zzx)$  /* line im(z)=0 */
 zzx1 : map(draw_format,zzx1)$  /* line z= 1   */
 zzx2 : map(draw_format,zzx2)$ /* line z = 2  */
 zzxm1 : map(draw_format,zzxm1)$  /* line z=-1  */
 zzxm2 : map(draw_format,zzxm2)$  /* line z=-2  */

 zzxhi   : map(draw_format,zzxhi)$  /* line im(z)=0 */
 zzx1hi  : map(draw_format,zzx1hi)$  /* line z= 1   */
 zzx2hi  : map(draw_format,zzx2hi)$ /* line z = 2  */
 zzxm1hi : map(draw_format,zzxm1hi)$  /* line z=-1  */
 zzxm2hi : map(draw_format,zzxm2hi)$  /* line z=-2  */

/* images of  z-lines */
 wwx  : map(draw_format,wwx)$  /* image of line im(z)=0 */
 wwx1 : map(draw_format,wwx1)$  /* line z= 1   */
 wwx2 : map(draw_format,wwx2)$ /* line z = 2  */
 wwxm1 : map(draw_format,wwxm1)$  /* line z=-1  */
 wwxm2 : map(draw_format,wwxm2)$  /* line z=-2  */
 
 /* points not a list of points */
 z0:draw_format(z0);
 w0:draw_format(w0);
 z0hi:draw_format(z0hi);
 /* zi:draw_format(zi); */
 wi:draw_format(wi);


print (" ------------------  draw ------------------------------------------------------ ")$

path:"~/maxima/batch/julia/parabolic/z2z/preimages/1/"$ /* pwd  */
 FileName:"i"$ /* without extension which is the terminal name */

 load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
 draw(
  terminal  = 'svg,
  file_name = concat(path,FileName),
  columns  = 3,
  dimensions=[1500,500], /*  x = y*columns  */
  
  gr2d(title = " z plane ",
   yrange = [-3,3],
   xrange = [-1,5],
   points_joined =true,
   grid = false,
   color         = red,
   point_size    = 0.2,
   point_type = filled_circle,
   points(zz),
   color = blue,
   points(zz1),
   color = green,
   points(zz2),
   color = gray,
   points(zzx),
   points(zzx1),
   points(zzx2),
   points(zzxm1),
   points(zzxm2),
   
   points_joined =false,
   color = black,
   point_size    = 0.8,
   key="origin",
   points([z0])
   
  ),

  gr2d(
   title = " w plane : w=h(z)= (z-1)/(z+1)",
   yrange = [-2.0,2.0],
   xrange = [-2.0,2.0],
   grid = false,
   xaxis       = false,
   
   points_joined =true,

   color         = red,
   point_size    = 0.2,
   point_type = filled_circle,
   points(ww),
   color = blue,
   points(ww1),
   color=green,
   points(ww2),

   color = gray,
   points(wwx),
   points(wwx1),
   points(wwx2),
   points(wwxm1),
   points(wwxm2),
   
   points_joined =false,
   color=black,
   point_size    = 0.8,
   key = "h(origin)",
   points([w0]),
   color = red,
   key="h(infinity)",
   points([wi])
),
   

  gr2d(title = " z plane : z=hi(w) = (1+w)/(1-w)",
   yrange = [-3,3],
   xrange = [-1,5],
   grid = false,
   xaxis       = false,
   
   points_joined =true,
   color         = red,
   point_size    = 0.2,
   point_type = filled_circle,
   points(zzhi),
   color = blue,
   points(zz1hi),
   color = green,
   points(zz2hi),   

   color = gray,
   points(zzxhi),
   points(zzx1hi),
   points(zzx2hi),
   points(zzxm1hi),
   points(zzxm2hi),
   
   point_size    = 0.8,
   points_joined =false,
   color = black,
   key = "hi(h(origin))",
   points([z0hi])
)

 );

1. ↑ Unit disk in wikipedia
2. ↑ Mapping half-plane to unit disk?
3. ↑ Adam Majewski: fatou-coordinate in maxima-cas