- #1

- 54

- 0

In complex analysis, what is understood by the

thank you

*multiplicity*of a pole?thank you

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Icosahedron
- Start date

- #1

- 54

- 0

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

thank you

thank you

- #2

- 64

- 0

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

- #3

mathwonk

Science Advisor

Homework Helper

- 11,269

- 1,468

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

- #4

- 64

- 0

May this be extended beyond polynomials?

- #5

- 54

- 0

thanks!

- #6

mathwonk

Science Advisor

Homework Helper

- 11,269

- 1,468

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.

- #7

- 64

- 0

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

- #8

HallsofIvy

Science Advisor

Homework Helper

- 41,847

- 967

[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.

- #9

- 64

- 0

That definition put everything in place, thanks!

Share: