Exercise 17.  Evaluate  ,  where C is the polygonal path from that consists of the
line segments from    and .  

Solution 17.

See text and/or instructor's solution manual.

Answer.   .  

Solution.  Use the polygonal path from  .   

Set up the parameterizations and compute the contour integral.

The function is      and the contour segments are   .

Find a parameterization of the polygonal path    from    consisting of the two line segments:

                    The curve    for  .  

                    The curve    for  .  

                    The contour  .  

                    Where    for  ,  and
                                  for  .    

Part (i).  A parameterization of the line segment from   ,  is

                    ,   for   ,   and
                    
                    ,  
                    .  

Along    we obtain  

                      

Part (ii).  A parameterization of the line segment from   ,  is

                    ,   for   ,   and
                    
                    ,  
                    .  

Along    we obtain  

                    

The function is      

and we add the contributions on the contour segments    and get

                       

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 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