Removable Singularity
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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a removable singularity of a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
is a point at which the function is
undefined Undefined may refer to: Mathematics * Undefined (mathematics), with several related meanings ** Indeterminate form, in calculus Computing * Undefined behavior, computer code whose behavior is not specified under certain conditions * Undefined ...
, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of that point. For instance, the (unnormalized) sinc function : \text(z) = \frac has a singularity at . This singularity can be removed by defining \text(0) := 1, which is the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of as tends to 0. The resulting function is holomorphic. In this case the problem was caused by being given an
indeterminate form In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this su ...
. Taking a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
expansion for \frac around the singular point shows that : \text(z) = \frac\left(\sum_^ \frac \right) = \sum_^ \frac = 1 - \frac + \frac - \frac + \cdots. Formally, if U \subset \mathbb C is an open subset of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
\mathbb C, a \in U a point of U, and f: U\setminus \ \rightarrow \mathbb C is a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
, then a is called a removable singularity for f if there exists a holomorphic function g: U \rightarrow \mathbb C which coincides with f on U\setminus \. We say f is holomorphically extendable over U if such a g exists.


Riemann's theorem

Riemann's theorem on removable singularities is as follows: The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at a is equivalent to it being analytic at a ( proof), i.e. having a power series representation. Define : h(z) = \begin (z - a)^2 f(z) & z \ne a ,\\ 0 & z = a . \end Clearly, ''h'' is holomorphic on D \setminus \, and there exists :h'(a)=\lim_\frac=\lim_(z - a) f(z)=0 by 4, hence ''h'' is holomorphic on ''D'' and has a Taylor series about ''a'': :h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, . We have ''c''0 = ''h''(''a'') = 0 and ''c''1 = ''h''(''a'') = 0; therefore :h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, . Hence, where ''z'' ≠ ''a'', we have: :f(z) = \frac = c_2 + c_3 (z - a) + \cdots \, . However, :g(z) = c_2 + c_3 (z - a) + \cdots \, . is holomorphic on ''D'', thus an extension of ''f''.


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)^f(z)=0. If so, a is called a
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
of f and the smallest such m is the order of a. So removable singularities are precisely the
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
s of order 0. A holomorphic function blows up uniformly near its other poles. #If an isolated singularity a of f is neither removable nor a pole, it is called an
essential singularity In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
. The Great Picard Theorem shows that such an f maps every punctured open neighborhood U \setminus \{a\} to the entire complex plane, with the possible exception of at most one point.


See also

*
Analytic capacity In the mathematical discipline of complex analysis, the analytic capacity of a compact subset ''K'' of the complex plane is a number that denotes "how big" a bounded analytic function on C \ ''K'' can become. Roughly speaking, ''γ''(''K ...
*
Removable discontinuity Continuous functions are of utmost importance in mathematics, functions and applications. However, not all Function (mathematics), functions are Continuous function, continuous. If a function is not continuous at a point in its Domain of a function ...


External links


Removable singular point
a
Encyclopedia of Mathematics
Analytic functions Meromorphic functions Bernhard Riemann