I feel as if I'm probably just being stupid, but I'm working through Griffiths' book on quantum mechanics and I can't seem to evaluate the following integral:
I(k) = $\int_{0}^{\infty}{(e^{(ik - a)x} + e^{-(ik + a)x})dx}$.
I've managed to get up to the following point:
I(k) = $\lim_{\beta \rightarrow \infty}[\frac{e^{(ik - a)\beta}}{ik - a} - \frac{e^{-(ik + a)\beta}}{ik + a}]_{0}^{\beta} = \lim_{\beta \rightarrow \infty} \frac{e^{(ik - a)\beta}}{ik - a} - 0 - \frac{1}{ik - a} + \frac{1}{ik + a}$,
but the solution manual goes straight to:
I(k) = $\lim_{\beta \rightarrow \infty}[\frac{e^{(ik - a)\beta}}{ik - a} - \frac{e^{-(ik + a)\beta}}{ik + a}]_{0}^{\beta} = -\frac{1}{ik - a} + \frac{1}{ik + a}$.
Assuming I haven't made any mistakes, the missing step seems to be the evaluation of the following limit:
$\lim_{\beta \rightarrow \infty} \frac{e^{(ik - a)\beta}}{ik - a}$
which presumably evaluates to 0, but I really cannot see how. I've been flicking through my old notes on complex functions and Riley/Hobson/Bence, but I still cannot see how this works. Any help at all would be appreciated.
This assumes $a>0$. Note that $|e^{r+is}|=e^r$ where $r,s\in \mathbb R$. In your case, $r=-a\beta$, so the result indeed follows.