Example 6.23.  Show that  [Graphics:Images/IntegralRepresentationMod_gr_46.gif],  where C is the circle  [Graphics:Images/IntegralRepresentationMod_gr_47.gif]  with positive orientation.

[Graphics:Images/IntegralRepresentationMod_gr_48.gif]

Explore Solution 6.23.

Enter the integrand  [Graphics:../Images/IntegralRepresentationMod_gr_53.gif]  and locate the singularities.

[Graphics:../Images/IntegralRepresentationMod_gr_54.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_55.gif]

 

 

Find the singularity that lie inside C.

[Graphics:../Images/IntegralRepresentationMod_gr_56.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_57.gif]

 

 

Thus,  [Graphics:../Images/IntegralRepresentationMod_gr_58.gif]  the only singularity that lies inside  [Graphics:../Images/IntegralRepresentationMod_gr_59.gif].  

The integrand f(z) is to be used is

[Graphics:../Images/IntegralRepresentationMod_gr_60.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_61.gif]

 

 

Use Cauchy's Integral Formula to evaluate the integral of  [Graphics:../Images/IntegralRepresentationMod_gr_62.gif] taken over C.

[Graphics:../Images/IntegralRepresentationMod_gr_63.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_64.gif]

 

 

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

[Graphics:../Images/IntegralRepresentationMod_gr_65.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_66.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_67.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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