Finding the zeros and residues of a conformal mapping function

Questions
,
1

miu[zeta_] := (epsilon[1](1 + zeta)^2 + epsilon[2](1 - zeta)^2)/(4*
zeta) + Sqrt[(epsilon[1](1 + zeta)^2 +
epsilon[2](1 - zeta)^2)^2 - 16zeta^2]/(4zeta)
omega[zeta_] := (a + b)/2miu[zeta] + (a - b)/(2miu[zeta])
omegaPoles = Solve[Denominator[omega[zeta]] == 0, zeta]

I’m attempting to find the poles and residue of the function omega[zeta]. However, the output for omegaPoles is empty. Are the poles of this function at zero or at infinity？

Thank you in advance for your help and suggestions.

2

Denominator works on the unsimplified expression. Use Together to get over a common denominator.

omegaPoles = Solve[Denominator[Together@omega[zeta]] == 0, zeta]
3

Thank you for your guidance. The output value is {{zeta → 0}}.