Complex analysis antiderivative existence

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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.

Homework Equations





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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Answers and Replies

  • #2
Dick
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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.

Homework Equations





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].
 

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