Removable singularity

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is ostensibly undefined, but, upon closer examination, the domain of the function can be enlarged to include the singularity (in such a way that the function remains holomorphic).

For instance, the function

:$f(z)\; =\; frac\{sin\; z\}\{z\}$

for "z" ≠ 0 has a singularity at "z" = 0. This singularity can be removed by defining "f"(0) = 1. The resulting function is a continuous, in fact holomorphic, function.

Formally, if "U" is an open subset of the complex plane C, "a" is a point of "U", and "f" : "U" - {"a"} → C is a holomorphic function, then "a" is called a removable singularity for "f" if there exists a holomorphic function "g" : "U" → C which coincides with "f" on "U" - {"a"}. We say "f" is holomorphically extendable over "a" if such a "g" exists.

Riemann's theorem

Riemann's theorem on removable singularities states when a singularity is removable:

Theorem. The following are equivalent:

:i) "f" is holomorphically extendable over "a".

:ii) "f" is continuously extendable over "a".

:iii) There exists a neighborhood of "a" on which "f" is bounded.

:iv) lim_{"z" → "a"}("z - a") "f"("z") = 0.

The implications i) ⇒ ii) ⇒ iii) ⇒ iv) are trivial. To prove iv) ⇒ i), we first recall that the holomorphy of a function at "a" is equivalent to it being analytic at "a", i.e. having a power series representation. Define

:

Then

:$h(z)\; -\; h(a)\; =\; (z\; -\; a)(z\; -\; a)f(z),\; ,$

where, by assumption, ("z - a")"f"("z") can be viewed as a continuous function on "D". In other words, "h" is holomorphic on "D" and has a Taylor series about "a":

:$h(z)\; =\; a\_2\; (z\; -\; a)^2\; +\; a\_3\; (z\; -\; a)^3\; +\; cdots\; .$

Therefore

:$g(z)\; =\; frac\{h(z)\}\{(z-a)^2\}$

is a holomorphic extension of "f" over "a", which proves the claim.

Other kinds of singularities

Unlike functions of a real variable, holomorphic functions are sufficiently rigid that their isolated singularities can be completely classified. A holomorphic function's singularity is either not really a singularity at all, i.e. a removable singularity, or one of the following two types:

#In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number "m" such that lim_{"z" → "a"}("z - a ")^"m+1""f"("z") = 0. If so, "a" is called a pole of "f" and the smallest such "m" is the order of "a". So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its poles.
#If an isolated singularity "a" of "f" is neither removable nor a pole, it is called an essential singularity. It can be shown that such an "f" maps every punctured open neighborhood "U" - {"a"} to the entire complex plane, with the possible exception of at most one point.

ee also

* Analytic capacity
* Removable discontinuity