- #1

sbh77

- 3

- 0

## Homework Statement

V(x) = \int \frac{d^3q}{(2\pi)^3} \frac{-g^2}{|\vec{q}|^2 + m^2} \exp^{i \vec{q} \cdot \vec{x}}

= -\frac{g^2}{4\pi^2} \int_0^{\infty} dq q^2 \frac{exp^{iqr}-exp^{-iqr}}{iqr} \frac{1}{q^2+m^2}

= \frac{-g^2}{4\pi^2 i r} \int_{-\infty}^{\infty} dq \frac{q exp^{iqr}}{q^2+m^2}

## Homework Equations

none

## The Attempt at a Solution

I understand how the measure changes to a single integral with a factor of 4\pi (integrating out the angles leaving the radial one) and I also know that a vector in spherical coordinates is represented just by a quantity in the radial direction. The problem I am having is I don't know where the term \frac{exp^{iqr}-exp^{-iqr}}{iqr} came from. It almost looks like sin but the "i" is in the denominator.

My last question is why does the second line become the third? I see the limits of integration are changed but why did the second exponential term disappear? Is it a symmetry thing?

Thanks!