Why does the function real return expressions containing i ?

For the following code, CEre is derived using the real function, but it contains i in the expression.

clc;clear all;
syms s;
syms  T1 T2 w tau1 tau2 real;
e1=(1-T1*s)/(1+T1*s);
e2=(1-T2*s)/(1+T2*s);
A=[0        1       0       0;
 0        0       1       0;
 0        0       0       1;
 -29.17   -56   -36.7   -10.1];
B1=[-1.55   1       0       0;
  -1    -0.3      0       0;
  0       0       0.5     0;
  -0.7    0       -0.34   -2.6];
B2=[0       0       0       0;
  1      1.5      4       0;
  0       0       0       0;
-0.33     0       0      -1.1];
B3=[0       0       0       0;
  0       0       0       0;
  0       0       0       0;
-0.08    -0.7     0       -1];
B4=zeros(4,4); B4(4,3)=-3;
n=size(A,1);
tau3=0.169; tau4=0.26;
cez=numden(simplify(det(w*1i*eye(n)-A-B1*e1-B2*e2-B3*(cos(tau3*w)-1i*sin(tau3*w))-B4*(cos(tau4*w)-1i*sin(tau4*w)))));
% ce=collect(cez,s);
cew=subs(cez,s,w*1i)
CEre=real(cew);
CEim=imag(cew);

After commenting off the function collect, the function real gives a result without the unity of complex i.

Answer 1

[Walter Roberson]

I have not tried the above code, but consider

syms w
real(w*1i)

The result may contain "i" because it is possible that w is complex itself.

In this particular case, the result could also be expressed as imag(w), but there are other cases that are more difficult to resolve, such as

real((w + 1i)^2)

[Qingbin]

syms  T1 T2 w tau1 tau2 real

In the code above, it is defined that w is real. So

real(w*1i) = 0

In fact, using the code above, all the variables are real.

Answer 2

[Ahmed A. Selman]

Check the math, e.g.:

...
  TheEq=w*1i*eye(n)-A-B1*e1-B2*e2- B3*(cos(tau3*w)-1i*sin(tau3*w))-B4*cos(tau4*w)-1i*sin(tau4*w); 
 TheDet=det(TheEq);
 TheSimple=simplify(TheDet);
 cez=numden(TheSimple);
...etc

Then see what do

   TheSimpleEq=simplify(TheEq);
   pretty(TheSimpleEq)

give.

Why do you use (1i) instead of (i) only? If it is not declared before, then (i or j) alone are assumed a unity of complex in Matlab.

[Walter Roberson]

1i is safer than i or j, as then you do not need to worry about i or j having been accidentally assigned.

[Ahmed A. Selman]

If someone wrote a code, he shouldn't have to worry about accidents :)

[Walter Roberson]

Be safe, program defensively. And make it easier for other people to read.