Why do you say that the given solutions are not real? Whether they are real or not depends on r. Do you have any assumptions for r that you forgot to include?

@Markiyan Hirnyk 

convert(2^(1/4)*exp((1/8)*(3*Pi*I)), RootOf):
allvalues(%):
expand(residue(z^2/(z^4+2*z^2+2)^2, z = %)):
evalf(%);

     0.117223193780180 - 0.0083307958818396*I

@Markiyan Hirnyk Replace roots(denom(Pf), z) by roots(denom(Pf)). However, Christian has already done the equivalent.

What's the point of me "seeing" the long sets of algebraic numbers? I already told you that the set contains the correct residue.

@Christian Wolinski 

roots(denom(Pf));

and

evala(Roots(denom(Pf)));

produce identical output. And that output has the roots expressed in terms of RootOfs.

@Christian Wolinski My _EnvExplicit is unevaluated. And the RootOfs are not all converted to radicals.

@Markiyan Hirnyk With the third version, the final set does contain the correct residue. The problem is that it contains three other numbers as well.

@Christian Wolinski This works:

roots(denom(Pf));

This doesn't work (it gives empty output):

roots(denom(Pf), z);

Based on ?roots, I don't understand why the latter doesn't work. According to my reading, the two should be equivalent. Here's the relevant help:

• The call roots(a) returns roots over the field implied by the coefficients present.  For example, if all the coefficients are rational, then the rational roots are computed.  If a has no roots in the implied coefficient field, then an empty list is returned.  This assumes that a is a univariate polynomials.•
• The call roots(a, K) computes the roots of a over the algebraic number field defined by K. Here K must be a single RootOf, or a list or set of RootOfs, or a single radical, or a list or set of radicals.  For example, if I is given as the second argument, then roots looks for the roots of a over the complex rationals.
• The calls roots(a, x) and roots(a, x, K) are equivalent to the above if a is univariate in x. Otherwise, it treats the other indeterminates in a as parameters, and finds all roots as above and ignoring symbolic roots.

@Markiyan Hirnyk For the sake of readability, the the last three lines of output are

           Pfs:=            []
           ANS:=           {}
           ANS2:= {}, k = 2^(1/4)*(-1)^(3/8)

@Markiyan Hirnyk What are the empty words? Did I accuse you of trespassing the copyright law? No.

So what does this undocumented command do?

@Markiyan Hirnyk If you want to confirm its existence and the existence of its copyright notice, then do

interface(verboseproc= 2):
print(`evala/toprof`);

or

showstat(`evala/toprof`);

See ?option and ?interface.

@Markiyan Hirnyk The error message is due to the non-existence of command split in modern Maple.

@Christian Wolinski You are obviously a master Maple coder and master algebraist. Why don't you get modern Maple? Become a beta tester and you can earn a copy for finding a few bugs. Otherwise, what would it take for you to get modern Maple?

@Markiyan Hirnyk The fact that it prints with the body elided (or redacted) proves that it has option `Copyright...`. See ?option.

@Christian Wolinski There is no command split in modern Maple. Perhaps you mean factor?