Path dependence (Complex Analysis)

[Matt100]

Are the integrals of the function f(z) = (1/(z-2) + (1/(z+1) + e^(1/z)

path independent in the following domain: {Rez>0}∖{2}

The domain is not simply connected

I know that path independence has 3 equivalent forms
that are

1) Integrals are independent if for every 2 points and 2 contours lying completely in the domain the integral along the first contour = the integral along the second contour

2) For every closed contour lying in the domain, the integral over that contour is 0 the integral over that contour is = 0

3) There exists a F(z) in the domain such that F'(z) = f(z) over the entire domain.

Which one of these can be used to show whether the integrals of f(z) = (1/(z-2) + (1/(z+1) + e^(1/z) are path dependent or not in the domain {Rez>0}∖{2}.

It seems like number 2 is the easiest to use but not sure how?

 

[Svein]

Are the integrals of the function f(z) = (1/(z-2) + (1/(z+1) + e^(1/z) path independent in the following domain: {Rez>0}∖{2}

No. If the integrals are path independent, the integral across the closed loop consisting of path₁ up and path₂ down must be 0. But the integral across any closed path that winds 1 or more times around z=2 is an integer multiple of 2πi.