Consider the function f defined over the region ohm C 0 fz

Consider the function f defined over the region ohm = C - {0} f(z) = 1/z, (z ohm) (a) Is ohm simply connected? (b) Let Gamma be the unit circle in ccd; whence, it is a simple closed curve in ccd that lies wholly in ohm. What is the value of the integral integral_Gamma 1/z dz? If ohm was simply connected what would you expect it to be?

Solution

Sigma is not simple conneted since a hole (origin) is there in the given region

(b) The integarl value is 2pi* i

If the sigma is simple connected then the integral value is Zero