My question is about the coefficients of a complex laurent series. As far as I know, there are three kinds of series: those which converge in a finite circular region around the expansion point z0,(aka taylor series), those that converge in a ring shaped region between two circles centered at z0, and those that converge for all points outside a circle centered at z0. For the first kind, I know that cn=0 for all n<0, because otherwise it would blow up at z0. My question is about the second and third cases.(adsbygoogle = window.adsbygoogle || []).push({});

Do there have to be both negative and positive n coefficients in the second case? I think so, because if there werent any negative n terms, what would stop you from shrinking the inner circle? and the same goes for positive n terms and expanding the outer circle. But I'm not sure.

For the third case, where the series converges for all z such that |z-z0| is bigger than some number, can there be any positve n terms? Again, it seems like no, because they would diverge if you got far enough out. But if the coefficients got small fast enough, say as 1/n!, then it might still work. For example, the e^x taylor series converges for all x, so it is sort of like this kind of a laurent series, and it has positive n terms.

Basically, my question is this: When can you assume the cn's are only nonzero for n>0, only for n<0, and when must you have both? If you have a bunch of singularites inside the contour your integrating over to find the coefficients, but you know there won't be an upper limit on |z-z0| for convergence, can you safely assume these singularities will all cancel out and leave 0 for n>0? If not always, when can you do this?

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# Coefficients of a complex laurent series

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