Example 6.25.  Show that[Graphics:Images/IntegralRepresentationMod_gr_107.gif],  where C is the circle  [Graphics:Images/IntegralRepresentationMod_gr_108.gif]  with positive orientation.

[Graphics:Images/IntegralRepresentationMod_gr_109.gif]

Explore Solution 6.25.

Enter the integrand  [Graphics:../Images/IntegralRepresentationMod_gr_114.gif]  and locate the singularities.

[Graphics:../Images/IntegralRepresentationMod_gr_115.gif]


[Graphics:../Images/IntegralRepresentationMod_gr_116.gif]

 

 

Find the singularity that lie inside  [Graphics:../Images/IntegralRepresentationMod_gr_117.gif].  

[Graphics:../Images/IntegralRepresentationMod_gr_118.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_119.gif]

 

 

Since  z = i  is a singularity of order  n = 4 , multiply the integrand by [Graphics:../Images/IntegralRepresentationMod_gr_120.gif] to obtain the function f(z).

[Graphics:../Images/IntegralRepresentationMod_gr_121.gif]


[Graphics:../Images/IntegralRepresentationMod_gr_122.gif]

 

 

Use Cauchy's Integral Formula for Derivatives to evaluate the integral of  [Graphics:../Images/IntegralRepresentationMod_gr_123.gif] taken over C.

[Graphics:../Images/IntegralRepresentationMod_gr_124.gif]

 

 

 

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_125.gif]

 

 

Thus, we have found the value of the contour integral.

[Graphics:../Images/IntegralRepresentationMod_gr_126.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_127.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_128.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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