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Partial answer : your second question is not legible, and the third doesn't make sanse without the second.

Maxima has a residue function :

(%i2) ? residue


 -- Function: residue (<expr>, <z>, <z_0>)

     Computes the residue in the complex plane of the expression <expr>
     when the variable <z> assumes the value <z_0>.  The residue is the
     coefficient of '(<z> - <z_0>)^(-1)' in the Laurent series for
     <expr>.

          (%i1) residue (s/(s**2+a**2), s, a*%i);
                                          1
          (%o1)                           -
                                          2
          (%i2) residue (sin(a*x)/x**4, x, 0);
                                           3
                                          a
          (%o2)                         - --
                                          6

(\%o2) \mathbf{true}

which can be called via the maxima_methods() interface:

sage: (1/x).maxima_methods().residue(x,0)
1
sage: (1/x).maxima_methods().residue(x,1)
0
sage: tan(x).maxima_methods().residue(x,pi/2)
-1
sage: tan(x).maxima_methods().residue(x,0)
0

HTH,