Newcomer to math stackexchange here.
Trying to understand an old engineering paper but the author kinda skimps on the details and I can't quite understand what they did and was hoping someone here could help me understand.
Paper in question: https://www.mathstat.dal.ca/~iron/math4190/Papers/traffic.pdf
Equations in question: going from (8) to (9) (pg 170-171)
$n$ is index for cars, $\omega$ is frequency, $t$ is time, $M$ is mass, $\lambda$ is some constant.
Equation (8) is
$$f_n = (1 + i \omega M /\lambda)^{-n} f_0$$
which combined with equation (6) $$u_n(t) = f_n e^{i \omega t}, f_0 = 1$$
supposedly gives me equation (9) $$u_n(t) = (1 +\omega^2 M^2 /\lambda^2)^{-n/2} exp(i[\omega t - n \cos^{-1}(1 +\omega^2 M^2 /\lambda^2)^{-1/2}])$$
So it seems that
$$f_n = (1 + i \omega M /\lambda)^{-n} = (1 +\omega^2 M^2 /\lambda^2)^{-n/2} exp(i[ - n \cos^{-1}(1 +\omega^2 M^2 /\lambda^2)^{-1/2}]) $$
but I have no clue how they derived that. The authors talk about it being "the usual Fourier analysis" of monochromatic components, but frankly I've never heard of monochromatic components before.
Any help or pointers would be greatly appreciated :/
Let's assume units have been chosen so that $M/\lambda=1$ just to economize on typesetting. The complex amplitude factor $A=\frac{ 1}{ 1+ i \omega}$ can be written in Cartesian/rectangular form as $A=\frac{1-i \omega}{1+ \omega^2}$ or in its equivalent Euler/polar form as $ A= r e^{i \phi}$ where $e^{i \phi} = \cos \phi + i \sin \phi$ and $r= |A|=\frac{1}{\sqrt{1+\omega^2}}$. From these we deduce $ \cos \phi = Re( A)/|A|= 1/|A| $ and thus $\phi = \arccos \frac{1}{ \sqrt{1+ \omega^2}}$. The polar description of $A$ is preferable for this author's application because he really wants to compute $A^n= r^n e^{i n \phi}$.
You may find it helpful to review phasor notation phasor