Is it Possible to Construct a Laurent Series of Sqrt(z) About Zero?

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The discussion centers on the feasibility of constructing a Laurent series for the function sqrt(z) around the point zero, which is identified as a branch point. Participants clarify that while a Laurent series typically converges on an annulus, sqrt(z) cannot be defined continuously in such a region due to its branch nature. Instead, a Puiseux series is suggested as an alternative representation for sqrt(z), although its effectiveness in complex analysis remains uncertain.

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Ancient_Nomad
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Hi,

My mathematics professor said that it is possible to construct a Laurent series of sqrt(z) about zero by integrating over a keyhole contour and then taking the limit R --> 0 where R radius of the inner circle. But I think he is mistaken. I don't understand how it is possible to have a Laurent series about zero, as it is a branch point.

Can someone please clarify this point, and tell me what the series is if such a series exists.

Also, then is it possible to have a laurent series for any function about its branch point by considering a similar contour.

Thanks.
 
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I think you're right; Laurent series converge on an annulus, and square root cannot be defined* on an annulus about the origin.

Square root can be expressed by a (rather boring) Puiseux series, but I'm not sure how well that works complex analytically.


*: I mean in a continuous way, of course.
 

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