Complex analysis antiderivative existence

Homework Statement

a) Does f(z)=1/z have an antiderivative over C/(0,0)?

b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1.

The Attempt at a Solution

a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least one closed contour the integral is nonzero. Thus f(z) cannot be path independent and thus cannot have an antiderivative over the domain.

b) By using the same reasoning it seems that no antiderivative over the domain exists either, but I'm not sure.

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Dick
Homework Helper

Homework Statement

a) Does f(z)=1/z have an antiderivative over C/(0,0)?

b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1.

The Attempt at a Solution

a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least one closed contour the integral is nonzero. Thus f(z) cannot be path independent and thus cannot have an antiderivative over the domain.

b) By using the same reasoning it seems that no antiderivative over the domain exists either, but I'm not sure.
For b) why don't you make sure by integrating over a contour? Try z=e^(i*t) for t in [0,2pi].