How to tell if a complex exponential blows up

Question

I'm following Griffiths' Introduction to Quantum Mechanics, where he's discussing the general solution to the delta-function potential problem. The solution he refers to is $$\psi(x)=Ae^{ikx}+Be^{-ikx}$$

Now, he says that neither of the the terms blows up. The case he is dealing with is $x<0$. I cannot see this, because it seems to me that the second term blows up if $x$ is negative. I suspect he is using some argument that involves the complex nature of the terms. I would like to know if indeed this is the case. If so, what is the argument to show that the terms do not blow up? Thanks.

Answer 1

$$\psi(x)=Ae^{ikx}+Be^{-ikx}$$

Therefore,

$$\psi(x)=A\cos({kx}) + iA\sin(kx)+B\cos({-kx}) + iB\sin(-kx)$$

To arrive at the above equation, I utilised the fact that $e^{\theta i} = \cos\theta + i\sin\theta$.

Now I can further simplify the above expression to give:

$$\psi(x)=A\cos({kx}) + iA\sin(kx)+B\cos({kx}) - iB\sin(kx)$$
Therefore, $$\psi(x)=(A+B)\cos({kx}) + i(A-B)\sin(kx)$$

Therefore $\psi$ is a complex function with an oscillating real part, and an oscillating imaginary part. Also, evidently, the function does not "blow up".

So in conclusion you are correct, it is due to the "i"s in the function that prevent it from blowing up, which would occur without the i.