My problem in short is an application of the following theorem, as found in:

Dragoslav S. Mitrinović and Jovan D. Kecić , The Cauchy Method of Residues , 1984 , D. Reidel Publishing Company, theorem 1, chapter 5.4.2, pages 184-185.

Suppose that the following conditions are satisfied:

1 The function f is holomorphic in the extended plane, except for in a finite amount of singularities.

2. On the interval (a,b) of the real axis f may have only simple poles.

Suppose that the following conditions are satisfied:

3. f has no singularities at {a, b}.

For the theorem in a latex enviroment, see the stack exchange link.

Are there any applications you know of? All I know of are fun demonstrations of how it works, but nothing 'real'.

What I primarily have in mind is within the areas of statistics or physics (theoretical or not).

This is a cross post across:

[Mathoverflow]

[PhysicsForums]

[Thephysicsforum]

[Statisticsforum]

[Mathhelpforum]

Edit: Regarding the links, I'll make sure they're up and running as soon as those at physicsforums realize that this isn't math solely.