Commandes à copier dans mathematica pour les figures de mon article .........

**************************************************************************************************************







bx[x_, y_, a_, R_, I_] := 
 I*(y - R)/x/
   Sqrt[(a + Abs[x])^2 + (y - R)^2]*(-EllipticK[
      4*a*Abs[x]/((a + Abs[x])^2 + (y - R)^2)] + (a^2 + 
        Abs[x]^2 + (y - R)^2)/((a - Abs[x])^2 + (y - R)^2)*
     EllipticE[4*a*Abs[x]/((a + Abs[x])^2 + (y - R)^2)])
**************************************************************************************************************
by[x_, y_, a_, R_, I_] := 
 I/Sqrt[(a + Abs[x])^2 + (y - R)^2]*(EllipticK[
     4*a*Abs[x]/((a + Abs[x])^2 + (y - R)^2)] + (a^2 - 
        Abs[x]^2 - (y - R)^2)/((a - Abs[x])^2 + (y - R)^2)*
     EllipticE[4*a*Abs[x]/((a + Abs[x])^2 + (y - R)^2)])
**************************************************************************************************************
StreamPlot[{bx[x, y, 1, 0, 1], by[x, y, 1, 0, 1]}, {x, -4, 4}, {y, -4,
   4}]

**************************************************************************************************************

StreamDensityPlot[{bx[x, y, 1, 0, 1], by[x, y, 1, 0, 1]}, {x, -4, 
  4}, {y, -4, 4}, ImageSize -> Large, StreamStyle -> Black, 
 ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]
**************************************************************************************************************

StreamDensityPlot[{bx[x, y, 1, 1, 1] + bx[x, y, 1, -1, 1], 
  by[x, y, 1, -1, 1] + by[x, y, 1, 1, 1]}, {x, -4, 4}, {y, -4, 4}, 
 ImageSize -> Large, StreamStyle -> Black, ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]
**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 1, 1] + bx[x, y, 1, -1, 1] + 
   bx[x, y, 1, 0, 1], 
  by[x, y, 1, -1, 1] + by[x, y, 1, 1, 1] + by[x, y, 1, 0, 1]}, {x, -4,
   4}, {y, -4, 4}, ImageSize -> Large, StreamStyle -> Black, 
 ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]
**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 0.5, 1] + bx[x, y, 1, -0.5, 1] + 
   bx[x, y, 1, 1.5, 1] + bx[x, y, 1, -1.5, 1], 
  by[x, y, 1, 0.5, 1] + by[x, y, 1, -0.5, 1] + by[x, y, 1, 1.5, 1] + 
   by[x, y, 1, -1.5, 1]}, {x, -4, 4}, {y, -4, 4}, ImageSize -> Large, 
 StreamStyle -> Black, ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]
**************************************************************************************************************

StreamDensityPlot[{bx[x, y, 1, 1, 1] + bx[x, y, 1, -1, 1] + 
   bx[x, y, 1, 2, 1] + bx[x, y, 1, -2, 1] + bx[x, y, 1, 0, 1], 
  by[x, y, 1, 1, 1] + by[x, y, 1, -1, 1] + by[x, y, 1, 2, 1] + 
   by[x, y, 1, -2, 1] + by[x, y, 1, 0, 1]}, {x, -4, 4}, {y, -4, 4}, 
 ImageSize -> Large, StreamStyle -> Black, ColorFunction -> Hue ,
  StreamPoints -> Fine]
**************************************************************************************************************

StreamDensityPlot[{bx[x, y, 1, 1.5, 1] + bx[x, y, 1, -1.5, 1], 
  by[x, y, 1, -1.5, 1] + by[x, y, 1, 1.5, 1]}, {x, -4, 4}, {y, -4, 4},
  ImageSize -> Large, StreamStyle -> Black, 
 ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]

**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 1, 1] + bx[x, y, 1, -1, 1], 
  by[x, y, 1, -1, 1] + by[x, y, 1, 1, 1]}, {x, -4, 4}, {y, -4, 4}, 
 ImageSize -> Large, StreamStyle -> Black, ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]
**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 0.5, 1] + bx[x, y, 1, -0.5, 1], 
  by[x, y, 1, -0.5, 1] + by[x, y, 1, 0.5, 1]}, {x, -4, 4}, {y, -4, 4},
  ImageSize -> Large, StreamStyle -> Black, 
 ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]

**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 0.25, 1] + bx[x, y, 1, -0.25, 1], 
  by[x, y, 1, -0.25, 1] + by[x, y, 1, 0.25, 1]}, {x, -4, 4}, {y, -4, 
  4}, ImageSize -> Large, StreamStyle -> Black, 
 ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]
**************************************************************************************************************

StreamDensityPlot[{bx[x, y, 1, 0.125, 5] + bx[x, y, 1, -0.125, 5], 
  by[x, y, 1, -0.125, 5] + by[x, y, 1, 0.125, 5]}, {x, -4, 4}, {y, -4,
   4}, ImageSize -> Large, StreamStyle -> Black, 
 ColorFunction -> "Rainbow" ,
  StreamPoints -> Fine]
**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 0.5, 1] + bx[x, y, 1, -0.5, -1], 
  by[x, y, 1, -0.5, -1] + by[x, y, 1, 0.5, 1]}, {x, -4, 4}, {y, -4, 
  4}, ImageSize -> Large, StreamStyle -> Black, ColorFunction -> Hue ,
  StreamPoints -> Fine]

**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 0.5, 4] + bx[x, y, 1, -0.5, 2] + 
   bx[x, y, 1, 1.5, 8] + bx[x, y, 1, -1.5, 1], 
  by[x, y, 1, 0.5, 4] + by[x, y, 1, -0.5, 2] + by[x, y, 1, 1.5, 8] + 
   by[x, y, 1, -1.5, 1]}, {x, -4, 4}, {y, -4, 4}, ImageSize -> Large, 
 StreamStyle -> Black, ColorFunction -> Hue ,
  StreamPoints -> Fine]

**************************************************************************************************************
StreamDensityPlot[{bx[x, y, 1, 0.5, 1] + bx[x, y, 0.5, -0.5, 1] + 
   bx[x, y, 2, 1.5, 1] + bx[x, y, 0.25, -1.5, 1], 
  by[x, y, 1, 0.5, 1] + by[x, y, 0.5, -0.5, 1] + by[x, y, 2, 1.5, 1] +
    by[x, y, 0.25, -1.5, 1]}, {x, -4, 4}, {y, -4, 4}, 
 ImageSize -> Large, StreamStyle -> Black, ColorFunction -> Hue ,
  StreamPoints -> Fine]
**************************************************************************************************************

StreamDensityPlot[{
  
  
  bx[x - 2, y, 0.5, 0, 1]
   - by[-y + 2, x, 0.5, 0, 1]
   - bx[x + 2, y, 0.5, 0, 1] +
   by[-y - 2, x, 0.5, 0, 1]
  
  
  ,
  by[x - 2, y, 0.5, 0, 1] +
   bx[-y + 2, x, 0.5, 0, 1]
   - by[x + 2, y, 0.5, 0, 1]
   - bx[-y - 2, x, 0.5, 0, 1]
  
  
  }, {x, -4, 4}, {y, -4, 4}, ImageSize -> Large, StreamStyle -> Black,
  ColorFunction -> Hue ,
  StreamPoints -> Fine]


**************************************************************************************************************

StreamDensityPlot[{
  
  
  bx[x - 2, y, 0.5, 0, 1]
   - by[-y + 2, x, 0.5, 0, 1]
   - bx[x + 2, y, 0.5, 0, 1] +
   by[-y - 2, x, 0.5, 0, 1] +
   
   bx[0.707*(x - 2*0.707) + 0.707*(y - 2*0.707), 
     0.707*(y - 2*0.707) - 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707 - 
   by[0.707*(x - 2*0.707) + 0.707*(y - 2*0.707), 
     0.707*(y - 2*0.707) - 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707 +
   -bx[-0.707*(x + 2*0.707) + 
       0.707*(y - 2*0.707), -0.707*(y - 2*0.707) - 
       0.707*(x + 2*0.707), 0.5, 0, 1]*0.707 - 
   by[-0.707*(x + 2*0.707) + 
      0.707*(y - 2*0.707), -0.707*(y - 2*0.707) - 0.707*(x + 2*0.707),
      0.5, 0, 1]*0.707 +
   -bx[-0.707*(x + 2*0.707) - 
       0.707*(y + 2*0.707), -0.707*(y + 2*0.707) + 
       0.707*(x + 2*0.707), 0.5, 0, 1]*0.707 + 
   by[-0.707*(x + 2*0.707) - 
      0.707*(y + 2*0.707), -0.707*(y + 2*0.707) + 0.707*(x + 2*0.707),
      0.5, 0, 1]*0.707 +
   bx[0.707*(x - 2*0.707) - 0.707*(y + 2*0.707), 
     0.707*(y + 2*0.707) + 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707 + 
   by[0.707*(x - 2*0.707) - 0.707*(y + 2*0.707), 
     0.707*(y + 2*0.707) + 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707
  
  ,
  by[x - 2, y, 0.5, 0, 1] +
   bx[-y + 2, x, 0.5, 0, 1]
   - by[x + 2, y, 0.5, 0, 1]
   - bx[-y - 2, x, 0.5, 0, 1] +
   bx[0.707*(x - 2*0.707) + 0.707*(y - 2*0.707), 
     0.707*(y - 2*0.707) - 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707 + 
   by[0.707*(x - 2*0.707) + 0.707*(y - 2*0.707), 
     0.707*(y - 2*0.707) - 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707 +
   bx[-0.707*(x + 2*0.707) + 
      0.707*(y - 2*0.707), -0.707*(y - 2*0.707) - 0.707*(x + 2*0.707),
      0.5, 0, 1]*0.707 - 
   by[-0.707*(x + 2*0.707) + 
      0.707*(y - 2*0.707), -0.707*(y - 2*0.707) - 0.707*(x + 2*0.707),
      0.5, 0, 1]*0.707 +
   -bx[-0.707*(x + 2*0.707) - 
       0.707*(y + 2*0.707), -0.707*(y + 2*0.707) + 
       0.707*(x + 2*0.707), 0.5, 0, 1]*0.707 - 
   by[-0.707*(x + 2*0.707) - 
      0.707*(y + 2*0.707), -0.707*(y + 2*0.707) + 0.707*(x + 2*0.707),
      0.5, 0, 1]*0.707 +
   -bx[0.707*(x - 2*0.707) - 0.707*(y + 2*0.707), 
      0.707*(y + 2*0.707) + 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707 + 
   by[0.707*(x - 2*0.707) - 0.707*(y + 2*0.707), 
     0.707*(y + 2*0.707) + 0.707*(x - 2*0.707), 0.5, 0, 1]*0.707
  
  
  }, {x, -4, 4}, {y, -4, 4}, ImageSize -> Large, StreamStyle -> Black,
  ColorFunction -> Hue ,
  StreamPoints -> Fine
 ]


**************************************************************************************************************