Example 8.12.    Evaluate  .  

Explore Solution 8.12.

Enter the function   and use substitution to obtain the complex function f[z].  

Locate the singularities of  .  

Find out which singularities lie inside the unit circle  .  

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

Mathematica  is capable of finding some of these difficult integrals.  The function    can be integrated.

However, direct substitution using and the endpoints of the interval seems to be erroneous.  This is due to the fact that in the interior of the interval, is undefined. Hence is discontinuous.

This is due to the fact that   is undefined.  A graph of    reveals that it has a discontinuity at    in the middle of the interval  .  This is a violation of the Fundamental Theorem of Calculus, which asserts that the integral of a continuous function must be continuous.

The indefinite integrand    can be used piecewise over    where it is continuous, by using the left and right hand limits at the endpoints.

The above answer is correct too.  Limits were required because    is not continuous at  .  

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