$\lim_{q\to\infty} \int_{0}^{\infty} e^{i(q+i\delta)r}-e^{-i(q-i\delta)r} dr$=?

Question

Find $\displaystyle \lim_{q\to\infty} \int_{0}^{\infty} e^{i(q+i\delta)r}-e^{-i(q-i\delta)r} dr$

I think this is similar to delta function $\delta(x-x')=\frac{1}{\pi}\int_{0}^{\infty}e^{ik(x-x')}dk$,but it seems that it doesn't work.So how can I deal with it? thanks

And the answer should be $\frac{1}{q+i\delta}+\frac{1}{q-i\delta}$

Answer 1

First of all the Dirac Delta integral is from $-\infty$ to $+\infty$.

This is avery trivial integral, it's nothing but

$$\int_0^{+\infty} e^{iqr - \delta r}\ \text{d}r$$

Which you can write as

$$\int_0^{+\infty} e^{-Ar}\ \text{d}r$$

Where $A = -iq + \delta$.

The solution is trivial.

Idem for the other part!