We will be interested in bound states namely, energy eigenstates that are normalizable. For this the energy E of the states must be negative. This is readily understood. If E > 0, any solutions in the region x > a where the potential vanishes would be a plane wave, extending all the way to infinity. Such a solution would not be normalizable.

I'm guessing that the requirement that bound states are energy eigenstates that are normalizable is by definition. I also get why E>0 leads to a solution of e^ikx, as given in earlier sections, which extends to infinity will not be normalizable and hence, won't be a bound state (by definition). But why is e^ikx a plane wave? On page 10, it looks as though a plane wave, represented at places with zero potential, is just an exponential decay, why do we have the complex i in the equation? Also, in wikipedia, it says that a plane wave is mathematically represented as a cosine or a sine, not an exponential. I don't understand what's happening here...