why maple does not evaluate this limit?

Question

When i type the following code maple does not evaluate the limit but it clearly exists.

> restart; 
> omega := proc (x, y) options operator, arrow; 2*[x, y, 1]/(1+x^2+y^2) end proc;

> phi := proc (x, y) options operator, arrow; Re((lambda*(x+I*y)+a+I*b)/(1-lambda*(a-I*b)*  (x+I*y))), Im((lambda*(x+I*y)+a+I*b)/(1-lambda*(a-I*b)*(x+I*y))) end proc;

> Omega := limit(omega(phi(x/(e^2*(x^2+y^2)), y/(e^2*(x^2+y^2)))), e = 0);

Thanks for helping me.

Answer 1

You appear to be attempting to map the various operations over a list, as evidenced by your use of [x,y,1] in the numerator of the expression in the body of omega.

But *, /, and limit will not automatically map over a list.

You can map the * and / by either using expand or the elementwise syntax *~ and /~. For taking the limit I use the map command below.

If you didn't intend to map operations over a list then please explain what you intended by [x,y,1].

Note also that the limit as e->0 can be obtained by Maple if various assumptions are made, or if certain "simplifications" (by evalc, which acts as if unknowns are real) are done prior to calling limit. By default Maple would otherwise consider the variables other than e as being complex.

restart; 

omega := (x, y) -> expand( 2*[x, y, 1]/(1+x^2+y^2) ):
#omega := (x, y) -> 2*~[x, y, 1]/~(1+x^2+y^2):

phi := (x, y) -> (Re((lambda*(x+I*y)+a+I*b)/(1-lambda*(a-I*b)*(x+I*y))),
                  Im((lambda*(x+I*y)+a+I*b)/(1-lambda*(a-I*b)*(x+I*y)))):

expr := omega(phi(x/(e^2*(x^2+y^2)), y/(e^2*(x^2+y^2)))):

map(limit,expr,e=0) assuming real;

      [                                / 2    2\]
      [      2 a            2 b      2 \a  + b /]
      [- -----------, - -----------, -----------]
      [   2    2         2    2       2    2    ]
      [  a  + b  + 1    a  + b  + 1  a  + b  + 1]

newexpr := evalc(expr):

map(limit,newexpr,e=0);

      [                                / 2    2\]
      [      2 a            2 b      2 \a  + b /]
      [- -----------, - -----------, -----------]
      [   2    2         2    2       2    2    ]
      [  a  + b  + 1    a  + b  + 1  a  + b  + 1]

Let us know, if you had something else in mind.