Mapping properties of some complex functions 

 

We illustrate the mapping properties by mapping a rectangular grid consisting of vertical and horizontal lines. No explanation concerning the Maple programming is given. 

> mymap:=proc(f,z,xmin,xmax,ymin,ymax,N)
local t,hx,hy,L,k;
hx:=(xmax-xmin)/N;
hy:=(ymax-ymin)/N;
L:=[];
for k from 0 to N do
L:=[op(L),plot([Re(f(xmin+hx*k+I*t)),Im(f(xmin+k*hx+I*t)),t=ymin..ymax],scaling=constrained,color=red)];
L:=[op(L),plot([Re(f(I*(ymin+hy*k)+t)),Im(f(I*(ymin+hy*k)+t)),t=xmin..xmax],scaling=constrained,color=green)];
end do:
L;
end proc:
 

>
 

> with(plots):
 

> myid:=z->z:
 

> A:=Array(1..2):
 

The first function is the square of a complex number: 

> f:=z->z^2;
 

proc (z) options operator, arrow; `*`(`^`(z, 2)) end proc
 

> A[1]:=display(mymap(myid,z,-2,2,0,2,16)):
 

> A[2]:=display(mymap(f,z,-2,2,0,2,16)):
 

> display(A);
 

Plot_2d Plot_2d

 

The next example is for the cosine. 

> gg:=z->cos(z);
 

proc (z) options operator, arrow; cos(z) end proc
 

> B:=Array(1..2):
 

> B[2]:=display(mymap(gg,z,-Pi,Pi,0,2,16)):
 

> B[1]:=display(mymap(myid,z,-Pi,Pi,0,2,16)):
 

> display(B);
 

Plot_2d Plot_2d

 

The exponential function in two different domains. 

> h:=z->exp(z);
 

proc (z) options operator, arrow; exp(z) end proc
 

> C:=Array(1..2):
 

> C[1]:=display(mymap(myid,z,-4,4,0,Pi,16)):
 

> C[2]:=display(mymap(h,z,-4,4,0,Pi,16)):
 

> display(C);
 

Plot_2d Plot_2d

 

> C[1]:=display(mymap(myid,z,-4,4,0,2*Pi,16)):
 

> C[2]:=display(mymap(h,z,-4,4,0,2*Pi,16)):
 

> display(C);
 

Plot_2d Plot_2d

 

The natural logarithm function 

> hh:=z->evalc(ln(z));
 

proc (z) options operator, arrow; evalc(ln(z)) end proc
 

> G:=Array(1..2):
 

> G[1]:=display(mymap(myid,z,0.1,4,0.1,4,16)):
 

> G[2]:=display(mymap(hh,z,0.1,4,0.1,4,16)):
 

> display(G);
 

Plot_2d Plot_2d

 

A polynomial of degree 3 

> f1:=z->z^3-3*z;
 

proc (z) options operator, arrow; `+`(`*`(`^`(z, 3)), `-`(`*`(3, `*`(z)))) end proc
 

> H:=Array(1..2):
 

> H[1]:=display(mymap(myid,z,0,2,0,2,16)):
 

> H[2]:=display(mymap(f1,z,0,2,0,2,16)):
 

> display(H);
 

Plot_2d Plot_2d

 

The inverse, with two different domains. 

> f2:=z->1/z;
 

proc (z) options operator, arrow; `/`(1, `*`(z)) end proc
 

> K:=Array(1..2):
 

> K[1]:=display(mymap(myid,z,0.25,4,0.25,4,16)):
 

> K[2]:=display(mymap(f2,z,0.25,4,0.25,4,16)):
 

> display(K);
 

Plot_2d Plot_2d

 

> L:=Array(1..2):
 

> L[1]:=display(mymap(myid,z,-4 ,4,-4,4,17)):
 

> L[2]:=display(mymap(f2,z,-4,4,-4,4,17)):
 

> display(L);
 

Plot_2d Plot_2d

 

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