7 (e).  Evaluate  ,  where C is the portion of    in the first quadrant.

Solution 7 (e).

See text and/or instructor's solution manual.

Answer.   .  

                    The contour      for   .

Solution Method I.  The function is    and the curve is     for    and we obtain

                       and  ,  
                     
                    then

                       

The last two real integrals are computed using

                      
                    
                    
                    and
                    

                      

We are done.  

Solution Method II.  We could use a method that uses the following complex computations.

                    The contour      for   .

The function is    and the curve is    for    and we obtain

                      and  ,  
                    
                    then

                       

Here we have used the calculations indicated by equation (6-8) in Section 6.1:

                       

We are really done.  

Aside.  After we have developed the Cauchy-Goursat Theorem in Section 6.3 and the Fundamental Theorem of Calculus in Section 6.4

we will be able to compute the integral in an easy fashion:

                    

We are really really done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell