Extra Example 1.  Evaluate the contour integrals  of    starting at the points   .  
(a)  Use the line segment joining the points.  
(b)  Use a portion of a parabola joining the points.

Explore Solution for Extra Example 1 (b)

(b)  Use the portion of a parabola connecting the points  ,  set up the parameterization and compute the contour integral.

Method (i).  The integral can be computed using the real and imaginary parts.

Method (ii).  The integral can be computed using a complex integrand.

Which is the same as the value along the other path because    is an analytic function

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