No its not complex exponential.
And it requires infinite energy because at x --> infinity, the function tends to infinity. It's energy density increases with x. So it need to be supplied with energy for infinity.
We know that a wave is represented by f(x - vt) and it follows the differential wave equation. e^(x - vt) satisfies both the condition. But is it really a wave? Because to sustain the wave we need infinite energy which is not possible. So what's happening here?