Why is it safe to assume point-wise voltage here is real-valued? 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.
 A: 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.
