Suppose we are given a domain D containing a simple closed contour C. Suppose that in addition we are given a point Zo with Zo NOT in D and
Explain why this ensures that D is not simply connected.
If C is contained in D, but point Zo is not in D then there are two options:
a) Zo is outside C and outside D, or
b) Zo is inside C, but in a region inside C that is not in D ( here i picture D as an annulus of two circles, C within the annulus)
For a) since Zo is outside C, f(z) = is analytic on and within C, and by the Cauchy-Goursat theorem
But since here it cannot be that Zo is outside C, hence option b) must b true.
Therefore it is the case that Zo is inside C, but in a region inside C that is not in D, which guarentees that D is not a simply connected domain.
I'm not convinced this is correct. I especially wonder if my assumption of only two cases a) and b) are correct.It makes sense to me but i feel i might be overlooking something. Would appreciate any comments!
Yes here C is the curve |z|=1, not sure why C was given, surely it is not necessary to know that C is |z|=1 to solve this problem ? But for the sake of including all information i retype exactly my problems as they are in the text.