Example 6.22.  Show that  [Graphics:Images/IntegralRepresentationMod_gr_27.gif],  where C is the circle  [Graphics:Images/IntegralRepresentationMod_gr_28.gif]  with positive orientation.

[Graphics:Images/IntegralRepresentationMod_gr_29.gif]

Explore Solution 6.22.

Enter the integrand  [Graphics:../Images/IntegralRepresentationMod_gr_32.gif]  and locate the singularities.

[Graphics:../Images/IntegralRepresentationMod_gr_33.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_34.gif]

 

 

Find the singularity that lie inside  [Graphics:../Images/IntegralRepresentationMod_gr_35.gif].  

[Graphics:../Images/IntegralRepresentationMod_gr_36.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_37.gif]

 

 

The integrand f(z) is to be used is

[Graphics:../Images/IntegralRepresentationMod_gr_38.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_39.gif]

 

 

Use Cauchy's Integral Formula to evaluate the integral of  [Graphics:../Images/IntegralRepresentationMod_gr_40.gif] taken over C.

[Graphics:../Images/IntegralRepresentationMod_gr_41.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_42.gif]

 

 

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

[Graphics:../Images/IntegralRepresentationMod_gr_43.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_44.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_45.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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