Applications of the residue theorem on integration

First we deduce Cauchy’s integral formula for simply connected domain D from the residue theorem:

Let f(z) be analytic  in a simply connected domain D  containing a simple closed curve C with a inside C.  Observe then that the function is analytic except at a where the residue is at ; that is, f(a).  By the residue theorem

or which is Cauchy’s integral formula.


Example.  Next we consider the example of  evaluating   by the residue theorem:  Let be large and be small. Now let represent the curve

Consider the integral where   The integrand is analytic in C except for a simple pole at i with residue =.

On the negative x-axis and so


 But and so and setting r=x in the first integral we obtain


as .  A similar argument with r replaced with R gives

as r.   

Consequently, letting and and equating the real parts gives .