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

[Ancient_Nomad]

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.

 

[Hurkyl]

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.