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Removable singularity

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Removable singularity

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.

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For instance, the (unnormalized) sinc function

sinc ( z ) = sin ⁡ z z

has a singularity at z = 0. This singularity can be removed by defining f(0) := 1, which is the limit of f as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by f being given an indeterminate form. Taking a power series expansion for sin ⁡ ( z ) z shows that

sinc ( z ) = 1 z ( ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! ) = ∑ k = 0 ∞ ( − 1 ) k z 2 k ( 2 k + 1 ) ! = 1 − z 2 3 ! + z 4 5 ! − z 6 7 ! + ⋯ .

Formally, if U ⊂ C is an open subset of the complex plane C , a ∈ U 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 U if such a g exists.

Riemann's theorem

Riemann's theorem on removable singularities is as follows:

Theorem. Let D ⊂ C be an open subset of the complex plane, a ∈ D a point of D and f a holomorphic function defined on the set D ∖ { a } . The following are equivalent:

  1. f is holomorphically extendable over a .
  2. f is continuously extendable over a .
  3. There exists a neighborhood of a on which f is bounded.
  4. lim z → a ( z − a ) f ( z ) = 0 .

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 ) = { ( z − a ) 2 f ( z ) z ≠ a , 0 z = a .

Clearly, h is holomorphic on D  {a}, and there exists

h ′ ( a ) = lim z → a ( z − a ) 2 f ( z ) − 0 z − a = lim z → a ( 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 ≠ 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 → 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 other poles.
  2. 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 ∖ { a } to the entire complex plane, with the possible exception of at most one point.

References

Removable singularity Wikipedia
(Text) CC BY-SA


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