During my research in Engineering, I came across this problem.
There is a complex valued function, with the domain as the imaginary axis.
\[P(j\omega) = P_R(\omega^2)+j\omega P_I(\omega^2)\]
I need to find the value of \(P(j\omega)\), at the sequence of points where it is real and draw appropriate conclusions.
There are two scenarios here. (1) Both \(P_R(x)\) and \(P_I(x)\) are polynomials in \(x\). (2) Not case 1.
For case 1 : As we are interested in values of \(P_R(\omega^2\)) at those \(\omega\), where \(P_I(\omega^2)=0\), it is obvious that we are interested ONLY in positive and real solutions to \(P_I(x)=0\).
My question to the community is this.
If there is no simpler solution, than finding all the roots; then the procedure which I envision would become more computationally complex than a conventional technique that is more than 80 years old.
If case 1 is worth pursuing, then case 2 could be an interesting extension.