6.24. Let denote
a fixed complex value. Show that, if C
is a simple closed positively oriented contour such
interior to C, then
, for any integer .
Explore Solution 6.24 (b).
that , when is
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