Question: without multiseries and selecting poles with positive imaginary parts

 Okay so I can see how to get the poles. But I have some further questions.

1) For beta AND Q both GREATER THAN 0 WHICH poles have positive imaginary part and is there a PROGRAMMATIC way of selecting these ones?

2) How do I calculate the residue at these poles? The approach given by Alec above seems useful and is much appreciated but I can't make it work. When I put the whole code into Maple at first there is an error message saying there is a missing semicolon but when I remove the tilde (squiggle ~) after op, convert and series I get the poles outputted but am told

  Error, `MultiSeries` does not evaluate to a module.

I am using Maple 9 and I am guessing multiseries is not available as when I input with(MultiSeries) Maple does not recognise the call.

with(MultiSeries);

Error, invalid input: with expects its 1st argument, pname, to be of type {package, module}, but received MultiSeries

So I tried using the Maple built in residue package as below for example using the first one of the poles but it will not evaluate.

s2 := (1/2/Pi+2*Q*u/(1+u^2))^2/((1+beta^2*(1/2/Pi+2*Q*u/(1+u^2))^2)*(1+u^2));

u6:= -2*beta*Q*Pi*(beta+2*I*Pi)/(4*Pi^2+beta^2)+(4*beta^4*Q^2*Pi^2+16*I*beta^3*Q^2*Pi^3-16*Pi^4*beta^2*Q^2-16*Pi^4-8*Pi^2*beta^2-beta^4)^(1/2)/(4*Pi^2+beta^2);

residue(s2,u=u6);

I guess hence the need for multiseries – on what version of maple did multiseries get introduced? So is there any other way of evaluating the residues?

I also tried using the formula for calculating residues and taking limits but this did not evaluate either.

> test5:=diff((u-u6)*s2, u);

> limit(test5,u=u6);

So in summary how do I select the poles with positive imaginary parts programmatically for beta and Q > 0 and how do I get the residues calculated?

Alec I notice you say that if beta>0, the 2nd, 3rd, and 6th poles have positive imaginary part but what if both beta and Q are greater than 0?