Example 6.24.  Let  [Graphics:Images/IntegralRepresentationMod_gr_80.gif]  denote a fixed complex value.  Show that, if C is a simple closed positively oriented contour such that  [Graphics:Images/IntegralRepresentationMod_gr_81.gif]  lies interior to C, then  

               [Graphics:Images/IntegralRepresentationMod_gr_82.gif],   and
(6-50)
               [Graphics:Images/IntegralRepresentationMod_gr_83.gif],   for any integer  [Graphics:Images/IntegralRepresentationMod_gr_84.gif]

[Graphics:Images/IntegralRepresentationMod_gr_85.gif]

Explore Solution 6.24 (a).

(a)   Show that [Graphics:../Images/IntegralRepresentationMod_gr_86.gif].  Use the Cauchy Integral Formula in the form [Graphics:../Images/IntegralRepresentationMod_gr_87.gif].  

[Graphics:../Images/IntegralRepresentationMod_gr_88.gif]


[Graphics:../Images/IntegralRepresentationMod_gr_89.gif]

 

 

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

[Graphics:../Images/IntegralRepresentationMod_gr_90.gif]




[Graphics:../Images/IntegralRepresentationMod_gr_91.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_92.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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