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, | 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>\lim_{z \rightarrow a} f(z) (z-a)^k = r \,</math> | ||
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 | 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 | |||
:<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]]. | ||
==References== | ==References== | ||
* {{cite book | author=Tom M. Apostol | title=Mathematical Analysis | edition=2nd ed | publisher=Addison-Wesley | year=1974 | pages=458 }} | * {{cite book | author=Tom M. Apostol | title=Mathematical Analysis | edition=2nd ed | publisher=Addison-Wesley | year=1974 | pages=458 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:00, 5 October 2024
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.