# Revision history [back]

### 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!