Path dependence (Complex Analysis)

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1. Apr 15, 2015

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?

2. Apr 15, 2015

Svein

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.