5.46 Understanding conformal mapping in maple

To plot mapping of complex function in maple, use [plots]conformal The trick is to how to specify the quadrant in the x-y plane. This example shows how.

Suppose we want to map the first quadrant. Then we specify the DIAGONAL points in the range, from the lower left corner to the upper right corner, which then should be 0..1+I Because 0 is the lower left corner, and \((1,i)\) is the upper right corner. Example:

restart; 
assume(y,real); 
assume(x,real); 
#f:= z->I+z*exp(I*Pi/4); 
f:= z->z^2; 
w:=f(x+I*y); 
u:=Re(w); 
v:=Im(w); 
plots:-conformal(f(z),z=0..1+I,grid=[16,16],numxy=[16,16],scaling=constrained);
 

This below uses the first TWO quadrant’s, i.e. the upper half of the x-y plane

restart; 
assume(y,real); 
assume(x,real); 
#f:= z->I+z*exp(I*Pi/4); 
f:= z->z^2; 
w:=f(x+I*y); 
u:=Re(w); 
v:=Im(w); 
plots:-conformal(f(z),z=-1-I..1+I,grid=[16,16],numxy=[16,16],scaling=constrained);
 

This below puts the plots next to each others so to see them

restart; 
assume(y,real); 
assume(x,real); 
f:= z->I+z*exp(I*Pi/4); 
#f:= z->z^2; 
w:=f(x+I*y); 
u:=Re(w); 
v:=Im(w); 
A := array(1..2): 
A[1]:=plots:-conformal(z,z=0..1+I/2,grid=[16,16],numxy=[16,16],scaling=constrained): 
A[2]:=plots:-conformal(f(z),z=0..1+I/2,grid=[16,16],numxy=[16,16],scaling=constrained): 
plots:-display(A);