(sigh) I know I've seen how to do this before, but I can't find the notes!

The problem is this:

$\displaystyle \int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta}$ where a is real.

It is simple enough to show that the integral is equal to a real number, so

$\displaystyle \int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta} = \int_0^{\pi} d \theta \, sin \theta \, cos(a \, sin \theta)$

as an alternate formula.

Comparing two lines in the book I got this from the integral should be equal to

$\displaystyle 2 \frac{sin \, a}{a}$

(For the case a = 0, the integral is simple and comes out to be equal to 2. Obviously I'm not worried about this case.)

Any takers? Thanks in advance.

-Dan