Example 8.24.  Evaluate  ,  where  .  

Explore Solution 8.24.

Enter the function and    and locate the isolated singularities.

Which poles lie in the upper half plane ?  

Compute the residues at  ,  and use the residue calculus to compute the value of the integral.

Further analysis must be done.  As shown in the text, the limit of the integrals taken over the semicircles are both zero.
Use the fact that when x is negative  .  Then use this parameterize the line segment on the negative x-axis and obtain  

.  

Equating the real parts in the above equation we obtain  

.  

which is the correct value when  a = 2.  Similar computations will establish the general result.

Aside. We can use Mathematica's built in "PrincipalValue" option to evaluate this definite integral with specific values of  a.

Which leads us to conjecture the general result  ,  where  .  

Aside. We can let Mathematica do the integration.

However, we might not be able to use the anti-derivative for the computation.

(c) 2006 John H. Mathews, Russell W. Howell