Removable singularity explained

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is 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 of that point.

For instance, the (unnormalized) sinc function, as defined by

sinc (z) = (\sin z)/(z) has a singularity at . This singularity can be removed by defining sinc ( 0 ) : = 1 , which is the limit of as tends to 0. The resulting function is holomorphic. In this case the problem was caused by being given an indeterminate form. Taking a power series expansion for $\backslash frac$ around the singular point shows thatsinc (z) = (1)/(z) \left(\sum ^{inf ty} _{k =0} (( -1 ) ^{kz} ^{2k+1})/(( 2 k +1 )!) \right) = \sum ^{inf ty} _{k =0} (( -1 ) ^{kz} ^{2k})/(( 2 k +1 )!) = 1 - (z ^{2})/(3 !) + (z ^{4})/(5 !) - (z ^{6})/(7 !) + … .

Formally, if

U \subset C is an open subset of the complex plane C , a \in U a point of U , and f: U\setminus \{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\setminus \{a\} . 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. Defineh(z) = \begin{cases} (z - a) ^{2} f(z) & z \ne a ,\\ 0 & z = a . \end{cases}

Clearly, h is holomorphic on

D \setminus \{a\} , and there existsh'(a) = \lim _{z\to} ((z - a) ^{2f(z)-0})/(z - a) = \lim _{z\to} (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} + … .

We have c₀ = h(a) = 0 and c₁ = h(a) = 0; therefore

h(z) = c _{2} (z - a) ^{2} + c _{3} (z - a) ^{3} + … .

Hence, where

z \ne a , we have:f(z) = (h(z))/((z - a) ^{2}) = c _{2} + c _{3} (z - a) + … .

However,

g(z) = c _{2} + c _{3} (z - a) + … .

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:

1. 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} (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 other poles.

1. If an isolated singularity

a of f is neither removable nor a pole, it is called an essential singularity. 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
• Removable discontinuity

External links

• Removable singular point at Encyclopedia of Mathematics