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. 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 has a singularity at z = 0. This singularity can be removed by defining sinc ( 0 ) := 1 {displaystyle { ext{sinc}}(0):=1} , which is the limit of sinc {displaystyle { ext{sinc}}} as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by sinc {displaystyle { ext{sinc}}} being given an indeterminate form. Taking a power series expansion for sin ⁡ ( z ) z {displaystyle {frac {sin(z)}{z}}} around the singular point shows that Formally, if U ⊂ C {displaystyle Usubset mathbb {C} } is an open subset of the complex plane C {displaystyle mathbb {C} } , a ∈ U {displaystyle ain U} a point of U {displaystyle U} , and f : U ∖ { a } → C {displaystyle f:Usetminus {a} ightarrow mathbb {C} } is a holomorphic function, then a {displaystyle a} is called a removable singularity for f {displaystyle f} if there exists a holomorphic function g : U → C {displaystyle g:U ightarrow mathbb {C} } which coincides with f {displaystyle f} on U ∖ { a } {displaystyle Usetminus {a}} . We say f {displaystyle f} is holomorphically extendable over U {displaystyle U} if such a g {displaystyle g} exists. Riemann's theorem on removable singularities is as follows: Theorem. Let D ⊂ C {displaystyle Dsubset mathbb {C} } be an open subset of the complex plane, a ∈ D {displaystyle ain D} a point of D {displaystyle D} and f {displaystyle f} a holomorphic function defined on the set D ∖ { a } {displaystyle Dsetminus {a}} . The following are equivalent: The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at a {displaystyle a} is equivalent to it being analytic at a {displaystyle a} (proof), i.e. having a power series representation. Define Clearly, h is holomorphic on D  {a}, and there exists by 4, hence h is holomorphic on D and has a Taylor series about a: