Isolated singularity: Difference between revisions
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In [[complex analysis]], an '''isolated singularity''' of a [[complex number|complex]]-valued [[function (mathematics)|function]] is a point at which the function is not [[holomorphic function|holomorphic]], but which has a [[neighbourhood]] on which the function is holomorphic. | In [[complex analysis]], an '''isolated singularity''' of a [[complex number|complex]]-valued [[function (mathematics)|function]] is a point at which the function is not [[holomorphic function|holomorphic]], but which has a [[neighbourhood]] on which the function is holomorphic. | ||
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* The [[absolute value]] of ''f'' is bounded on ''N''; in this case ''f'' tends to a limit at ''a'', and the singularity is [[removable singularity|removable]]. | * The [[absolute value]] of ''f'' is bounded on ''N''; in this case ''f'' tends to a limit at ''a'', and the singularity is [[removable singularity|removable]]. | ||
* The absolute value |''f''| tends to infinity as ''f'' tends to ''a''; in this case some power of ''z''-''a'' times ''f'' is bounded, and the singularity is a [[pole (complex analysis)|pole]]. | * The absolute value |''f''| tends to infinity as ''f'' tends to ''a''; in this case some power of ''z''-''a'' times ''f'' is bounded, and the singularity is a [[pole (complex analysis)|pole]]. | ||
* Neither of the above occurs, and the singularity is [[isolated essential singularity|essential]]. | * Neither of the above occurs, and the singularity is [[isolated essential singularity|essential]].[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:01, 3 September 2024
In complex analysis, an isolated singularity of a complex-valued function is a point at which the function is not holomorphic, but which has a neighbourhood on which the function is holomorphic.
Suppose that f is holomorphic on a neighbourhood N of a except possibly at a. The behaviour of the function can be of one of three types:
- The absolute value of f is bounded on N; in this case f tends to a limit at a, and the singularity is removable.
- The absolute value |f| tends to infinity as f tends to a; in this case some power of z-a times f is bounded, and the singularity is a pole.
- Neither of the above occurs, and the singularity is essential.