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Please see my EMFT online notes.

Where it says I assumed $\hat{V}(x)$ to be real-valued out of laziness. But down lower where a general solution for $\hat{V}(x)$ is given, it appears that $V(x)$ could take on complex values for some coefficients and points $x$ along the transmission line. The book I'm using just jumps right over this as if it's obvious, and doesn't even say whether it should be real or complex-valued.

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Is this a math question or a physics question? – Jesse Madnick Jul 1 '12 at 6:40
Math. The goal is to find a derivation of first order ODE when you assume that $\hat{V}, \hat{I}$ are $\mathbb{C}$-valued. – Enjoys Math Jul 1 '12 at 16:01

I think I've found the meat of the problem. So there are these differential equations in Electromagnetics for modelling a long transmission line or strip of circuit board copper. They're given by $\partial_x v = R i + L \partial_t i$ together with its electric dual. In order to see what an alternating voltage and current looks like when put on the line and also to observe the "incident + reflected" wave property (this is called Sinusoidal Steady State), the form of $v(x,t)$ and $i(x,t)$ is restricted to $v(x,t) = Re\{\hat{V}(x) e^{j \omega t} \}$ and similarly with $i(x,t)$. You may be wondering how both current and voltage can have a sine wave with time with the same phase, however, notice that the possible difference in phase can be included in $\hat{V}(x)$. Using these forms for current and voltage, the work needed to solve the PDE is transformed into solving two ODEs in terms of $\hat{V}$ and $\hat{I}$. The ODEs are linear and are of the form $\frac{d\hat{V}(x)}{dx} = Z \hat{I}(x)$ together with dual.

If we assume $\hat{V}(x)$ to be $\mathbb{R}$-valued, then the derivation of those two ODEs is simple. But later we run into the problem that the general solution of those ODEs is $\mathbb{C}$-valued. So we must assume that they are complex valued. Since the ODEs are linear, a complex solution gives two real solutions to the PDEs (the components of the complex solution), and if $v, i$ are solutions of the Sinusoidal Steady State form, then they give two complex solutions to the ODEs: $\bar{v} = v + Im\{\hat{V}(x)e^{j \omega t}\} j$ and similar for $\hat{I}$.

So there is some type of 1-to-1 correspondence between complex solutions of the ODEs to real solutions of the PDEs. In other words, it's safe to solve the ODEs for complex solutions of the form $\hat{V}(x) e^{j \omega t}$, and take the real part to get a real solution to the PDEs, and this will cover all possible solutions for $v$ of the given form.

Add I put this small sequence of iff statements on my site, so might as well copy it here: $$ d_x \hat{V}(x) = Z \hat{I}(x) \\ \partial_x (\hat{V}(x) e^{j \omega t}) = Z \hat{I}(x) e^{j\omega t} \\ lhs = (R + j \omega L) \hat{I}(x) e^{j\omega t} \\ lhs = (R + L \partial_t) (\hat{I}(x) e^{j\omega t}) \\ $$ where the last line is a few steps away from being the telegrapher PDE equations. So the math works out similarly to the math with only one independent variable $t$ that you see when studying "phasors" in circuit analysis.

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