Example 6.24.  Let    denote a fixed complex value.  Show that, if C is a simple closed positively oriented contour such that    lies interior to C, then  

               ,   and
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               ,   for any integer  . 

Explore Solution 6.24 (b).

(b)   Show that  ,  when    is an integer.
Remark.  If  m  is an integer and  m < 0,  then  m = -n,  where  n  is a positive integer and the integrand is  
which is a polynomial of degree n.  Hence in this case, f(z) would be analytic and the Cauchy-Goursat theorem implies that the value of the integral is zero.

Now we consider the case when m is a positive integer and  m > 1.
Use the Cauchy's Integral Formulae for Derivatives in the form  

For illustration.  We use m = 5,  and  f(z) = 1.  Then  m = 5 = n + 1 implies that  n = 4.

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

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