Integrals of the function f(z) = e^(1/z) (complex analysis)

[Matt100]

How do you integrate f(z) = e^(1/z) in the multiply connected domain {Rez>0}∖{2}

It seems like integrals of this function are path independent in this domain since integrals of e^(1/z) exist everywhere in the domain {Rez>0}∖{2}. Is that correct?

 

[mfb]

I agree.
Where is the point in excluding that point, by the way? It is not special.

 

[Svein]

I seem to remember that the function given by f(z)=0 for Re(z)≤0 and $f(z)=e^{-\frac{1}{z}}$ for Re(z)>0 is ℂ^∞ in the whole complex plane, but not analytic...

 

[mfb]

Why?
The Laurent series is easy to construct here.

 

[Svein]

I seem to remember that the function given by f(z)=0 for Re(z)≤0 and $f(z)=e^{-\frac{1}{z}}$ for Re(z)>0 is ℂ^∞ in the whole complex plane, but not analytic...

Sorry, that should be C^∞.