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## Main Question or Discussion Point

In complex analysis, what is understood by the

thank you

*multiplicity*of a pole?thank you

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In complex analysis, what is understood by the *multiplicity* of a pole?

thank you

thank you

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For [tex]f(z)=\frac{1}{(z-z_0)^n}[/tex], the pole at [tex]z=z_0[/tex] has multiplicity n

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mathwonk

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i.e. the order of the pole of f at z, equals the multiplicity of the zero of 1/f, at z.

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May this be extended beyond polynomials?

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thanks!

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mathwonk

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on any open set, the fraction field of the holomorphic functions form what is called the field of meromorphic functions on that set. those have at worst poles as non holomorphic points.

e^(1/z) has a worse than pole point at z = 0. isolated non holomorphic points are called (isolated) singularities. the simplest actual singularities of functions defined by power series, possibly infinite in both directions, i.e. summed over all integer powers of z, are the poles.

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Does this also mean that an isolated/(essential?) singularity is a pole with infinite multiplicity?

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HallsofIvy

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[itex]\lim_{z\rightarrow a} (z-a)^nf(z)[/itex] exists but [itex]\lim_{z\rightarrow a} (z-a)^{n-1}f(z)[/itex] does not.

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That definition put everything in place, thanks!

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