Example 6.21.  Show that  [Graphics:Images/IntegralRepresentationMod_gr_5.gif],  where C is the circle  [Graphics:Images/IntegralRepresentationMod_gr_6.gif]  with positive orientation.

[Graphics:Images/IntegralRepresentationMod_gr_7.gif]

Explore Solution 6.21.

Enter the integrand  [Graphics:../Images/IntegralRepresentationMod_gr_13.gif]  and locate the singularities.

[Graphics:../Images/IntegralRepresentationMod_gr_14.gif]


[Graphics:../Images/IntegralRepresentationMod_gr_15.gif]

 

 

Find the singularity that lie inside  [Graphics:../Images/IntegralRepresentationMod_gr_16.gif].  

[Graphics:../Images/IntegralRepresentationMod_gr_17.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_18.gif]

 

 

The function f(z) in the numerator

[Graphics:../Images/IntegralRepresentationMod_gr_19.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_20.gif]

 

 

Use Cauchy's Integral Formula to evaluate the integral of  [Graphics:../Images/IntegralRepresentationMod_gr_21.gif] taken over C.

[Graphics:../Images/IntegralRepresentationMod_gr_22.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_23.gif]

 

 

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

[Graphics:../Images/IntegralRepresentationMod_gr_24.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_25.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_26.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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