Removable singularity: Difference between revisions
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In [[complex analysis]], a '''removable singularity''' is a type of [[singularity]] of a [[function (mathematics)|function]] of a [[complex number|complex]] variable which may be removed by redefining the function value at that point. | In [[complex analysis]], a '''removable singularity''' is a type of [[singularity]] of a [[function (mathematics)|function]] of a [[complex number|complex]] variable which may be removed by redefining the function value at that point. | ||
A function ''f'' has a removable singularity at a point ''a'' | A function ''f'' has a removable singularity at a point ''a'' if there is a neighbourhood of ''a'' in which ''f'' is [[holomorphic function|holomorphic]] except at ''a'' and the limit <math>\lim_{z \rightarrow a} f(z)</math> exists. | ||
In this case, defining the value of ''f'' at ''a'' to be equal to this limit (which makes ''f'' continuous at ''a'') gives a function holomorphic in the whole neighbourhood. | In this case, defining the value of ''f'' at ''a'' to be equal to this limit (which makes ''f'' continuous at ''a'') gives a function holomorphic in the whole neighbourhood. | ||
An isolated singularity may be either removable, a [[pole (complex analysis)|pole]], or an [[essential singularity]]. | An isolated singularity may be either removable, a [[pole (complex analysis)|pole]], or an [[essential singularity]]. | ||
==References== | |||
* {{cite book | author=Tom M. Apostol | title=Mathematical Analysis | edition=2nd ed | publisher=Addison-Wesley | year=1974 | pages=458 }} | |||
Latest revision as of 02:29, 25 October 2013
In complex analysis, a removable singularity is a type of singularity of a function of a complex variable which may be removed by redefining the function value at that point.
A function f has a removable singularity at a point a if there is a neighbourhood of a in which f is holomorphic except at a and the limit exists. In this case, defining the value of f at a to be equal to this limit (which makes f continuous at a) gives a function holomorphic in the whole neighbourhood.
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.