Pole (complex analysis): Difference between revisions

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In [[complex analysis]], a '''pole''' is a type of [[singularity]] of a [[function (mathematics)|function]] of a [[complex number|complex]] variable.  In the neighbourhood of a pole, the function behave like a negative power.
In [[complex analysis]], a '''pole''' is a type of [[singularity]] of a [[function (mathematics)|function]] of a [[complex number|complex]] variable.  In the neighbourhood of a pole, the function behave like a negative power.


A function ''f'' has a pole of order ''k'', where ''k'' is a positive integer, with (non-zero) residue ''r'' at a point ''a'' if the limit
A function ''f'' has a pole of order ''k'', where ''k'' is a positive integer, at a point ''a'' if the limit


:<math>\lim_{z \rightarrow a} f(z) (z-a)^k = r . \,</math>.
:<math>\lim_{z \rightarrow a} f(z) (z-a)^k = r \,</math>
 
for some non-zero value of ''r''.


The pole is an ''[[isolated singularity]]'' if there is a neighbourhood of ''a'' in which ''f'' is [[holomorphic function|holomorphic]] except at ''a''.  In this case the function has a [[Laurent series]] in a neighbourhood of ''a'', so that ''f'' is expressible as a power series
The pole is an ''[[isolated singularity]]'' if there is a neighbourhood of ''a'' in which ''f'' is [[holomorphic function|holomorphic]] except at ''a''.  In this case the function has a [[Laurent series]] in a neighbourhood of ''a'', so that ''f'' is expressible as a power series
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:<math> f(z) = \sum_{n=-k}^\infty c_n (z-a)^n , \,</math>
:<math> f(z) = \sum_{n=-k}^\infty c_n (z-a)^n , \,</math>


where the leading coefficient <math>c_{-k} = r</math>.
where the leading coefficient <math>c_{-k} = r</math>.  The [[residue (mathematics)|residue]] of ''f'' is the coefficient <math>c_{-1}</math>.


An isolated singularity may be either [[removable singularity|removable]], a pole, or an [[essential singularity]].
An isolated singularity may be either [[removable singularity|removable]], a pole, or an [[essential singularity]].

Revision as of 15:56, 12 November 2008

In complex analysis, a pole is a type of singularity of a function of a complex variable. In the neighbourhood of a pole, the function behave like a negative power.

A function f has a pole of order k, where k is a positive integer, at a point a if the limit

for some non-zero value of r.

The pole is an isolated singularity if there is a neighbourhood of a in which f is holomorphic except at a. In this case the function has a Laurent series in a neighbourhood of a, so that f is expressible as a power series

where the leading coefficient . The residue of f is the coefficient .

An isolated singularity may be either removable, a pole, or an essential singularity.

References

  • Tom M. Apostol (1974). Mathematical Analysis, 2nd ed. Addison-Wesley, 458.