# Imaginary part of a ratio of polynomials

When working with polynomial rings with complex coefficients, how to pass the assumption that the unknown variable belongs to the Reals set?

I am trying to recover and plot the imaginary part of the the ratio of polynomials *arg2*, but as sage assumes the independent variable is complex, it takes a long time and returns a huge answer. But I think the answer could be much simpler if it assumes that the independent variable belongs to the Reals set. The *assume()* function doesn't seems to work in this case.

*arg* has complex terms in the numerator and denominator. *arg2* has complex terms only on the numerator. So, it should be easy to obtain the imaginary part of *arg2*.

```
s = polygen(CC, "s")
omega = polygen(CC, "omega")
I = CC(I)
# assume(omega, "real") # this does not work
s21 = 0.894292331092066/(s^4 + 2.14081977623463*s^3 + 3.15237117897600*s^2 + 2.31898630138664*s + 0.902488008823108)
s21_omega = s21.subs(s=I*omega)
s21_omega_diff = s21_omega.derivative()
arg = 1/s21_omega*s21_omega_diff
num = arg.numerator()
den = arg.denominator()
den_conj = den.map_coefficients(lambda z: z.conjugate())
num2 = (num*den_conj).map_coefficients(lambda z: 0 if abs(z.real())<1e-10 and abs(z.imag())<1e-10 else z.imag()*I if abs(z.real())<1e-10 and abs(z.imag())>=1e-10 else z.real() if abs(z.real())>=1e-10 and abs(z.imag())<1e-10 else z)
den2 = (den*den_conj).map_coefficients(lambda z: 0 if abs(z.real())<1e-10 and abs(z.imag())<1e-10 else z.imag()*I if abs(z.real())<1e-10 and abs(z.imag())>=1e-10 else z.real() if abs(z.real())>=1e-10 and abs(z.imag())<1e-10 else z)
arg2 = num2/den2
print(arg2)
print(imag(arg2))
# plot(-imag(arg2), (-1.5, 1.5)).show()
```

Output:

```
(-3.19903509380033*omega^15 - 1.71213939841908*I*omega^14 + 9.63823791915898*omega^13 + 3.11426578979509*I*omega^12 - 15.8129908994689*omega^11 - 4.60245136409720*I*omega^10 + 13.7326241083798*omega^9 + 1.24770230131999*I*omega^8 - 9.58497293293397*omega^7 - 0.760732566992820*I*omega^6 + 4.72291963120453*omega^5 - 2.52135706735033*I*omega^4 - 2.44038907746050*omega^3 - 0.463630243025939*I*omega^2 + 0.203401406783201*omega - 1.36326886734867*I)/(0.799758773450081*omega^16 - 2.75378226261685*omega^14 + 5.27099696648963*omega^12 - 5.49304964335183*omega^10 + 4.79248646646703*omega^8 - 3.14861308746971*omega^6 + 2.44038907746050*omega^4 - 0.406802813566401*omega^2 + 0.530548112702672)
2.55845638284151*imag_part(omega)^31/(0.639614095710378*imag_part(omega)^32 + 10.2338255313661*imag_part(omega)^30*real_part(omega)^2 + 76.7536914852462*imag_part(omega)^28*real_part(omega)^4 + 358.183893597743*imag_part(omega)^26*real_part(omega)^6 + 1164.09765419317*imag_part(omega)^24*real_part(omega)^8 + 2793.83437005430*imag_part(omega)^22*real_part(omega)^10 + 5122.02967846394*imag_part(omega)^20*real_part(omega)^12 + 7317.18525490165*imag_part(omega)^18*real_part(omega)^14 + 8231.83341181278*imag_part(omega)^16*real_part(omega)^16 + 7317.18525493145*imag_part(omega)^14*real_part(omega)^18 + 5122.02967846394*imag_part(omega)^12*real_part(omega)^20 + 2793.83437005430*imag_part(omega)^10*real_part(omega)^22 + 1164.09765419317*imag_part(omega)^8*real_part(omega)^24 + 358.183893597743*imag_part(omega)^6*real_part(omega)^26 + ... (huge answer)
```

Thank you!