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Laurent series

principal part
Laurent expansion
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Mathematics Subject Classification

30B10 no label found


I wonder if the series necessarily diverges if |z-a| < R_1 or |z-a| > R_2 ?

Yes, for the same reason a taylor series necessarily diverges outside of the radius of convergence R.

Past R_2, the laurent series must diverge since the positive-power Taylor tail of the series diverges. Inside R_1, the laurent series must diverge since the negative-power Laurent head of the series diverges (which you can see by inverting z to 1/z and inverting the radius R_1 as well, and treating the result as a Taylor series).

Behavior on the boundary of the annulus is unpredictable in general.

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