Exercise 5.  Evaluate    from    along the following contours, as shown in Figures 6.13 (a)  and  6.13 (b).  

5 (a).  The polygonal path    with vertices  ,   as shown in Figures 6.13 (a).

                         Figure 6.13 (a).  The contour  .  

Solution 5 (a).

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 three line segments:

                    The curve    for  .  

                    The curve    for  .  

                    The curve    for  .  

                    Figure 6.13 (a).  The contour  .  

                    Where    for  ,  and
                                  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  

                    

Part (iii).  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

                       

This solution is complements of the authors.

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