Removable singularity

Removable singularity   (Redirected from Riemann's theorem on removable singularities) Jump to navigation Jump to search This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Removable singularity" – news · newspapers · books · scholar · JSTOR (July 2021) (Learn how and when to remove this template message) A graph of a parabola with a removable singularity at x = 2 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 {displaystyle {text{sinc}}(z)={frac {sin z}{z}}} has a singularity at z = 0. This singularity can be removed by defining {displaystyle {text{sinc}}(0):=1,} which is the limit of sinc as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by sinc being given an indeterminate form. Taking a power series expansion for {textstyle {frac {sin(z)}{z}}} around the singular point shows that {displaystyle {text{sinc}}(z)={frac {1}{z}}left(sum _{k=0}^{infty }{frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}right)=sum _{k=0}^{infty }{frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{frac {z^{2}}{3!}}+{frac {z^{4}}{5!}}-{frac {z^{6}}{7!}}+cdots .} Formally, if {displaystyle Usubset mathbb {C} } is an open subset of the complex plane {displaystyle mathbb {C} } , {displaystyle ain U} a point of {displaystyle U} , and {displaystyle f:Usetminus {a}rightarrow mathbb {C} } is a holomorphic function, then {displaystyle a} is called a removable singularity for {displaystyle f} if there exists a holomorphic function {displaystyle g:Urightarrow mathbb {C} } which coincides with {displaystyle f} on {displaystyle Usetminus {a}} . We say {displaystyle f} is holomorphically extendable over {displaystyle U} if such a {displaystyle g} exists.

Contents 1 Riemann's theorem 2 Other kinds of singularities 3 See also 4 External links Riemann's theorem Riemann's theorem on removable singularities is as follows: Theorem —  Let {displaystyle Dsubset mathbb {C} } be an open subset of the complex plane, {displaystyle ain D} a point of {displaystyle D} and {displaystyle f} a holomorphic function defined on the set {displaystyle Dsetminus {a}} . The following are equivalent: {displaystyle f} is holomorphically extendable over {displaystyle a} . {displaystyle f} is continuously extendable over {displaystyle a} . There exists a neighborhood of {displaystyle a} on which {displaystyle f} is bounded. {displaystyle lim _{zto 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 {displaystyle a} is equivalent to it being analytic at {displaystyle a} (proof), i.e. having a power series representation. Define {displaystyle h(z)={begin{cases}(z-a)^{2}f(z)&zneq a,\0&z=a.end{cases}}} Clearly, h is holomorphic on {displaystyle Dsetminus {a}} , and there exists {displaystyle h'(a)=lim _{zto a}{frac {(z-a)^{2}f(z)-0}{z-a}}=lim _{zto a}(z-a)f(z)=0} by 4, hence h is holomorphic on D and has a Taylor series about a: {displaystyle h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+cdots ,.} We have c0 = h(a) = 0 and c1 = h'(a) = 0; therefore {displaystyle h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+cdots ,.} Hence, where z ≠ a, we have: {displaystyle f(z)={frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+cdots ,.} However, {displaystyle 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 {displaystyle m} such that {displaystyle lim _{zrightarrow a}(z-a)^{m+1}f(z)=0} . If so, {displaystyle a} is called a pole of {displaystyle f} and the smallest such {displaystyle m} is the order of {displaystyle a} . So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its other poles. If an isolated singularity {displaystyle a} of {displaystyle f} is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an {displaystyle f} maps every punctured open neighborhood {displaystyle Usetminus {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 Categories: Analytic functionsMeromorphic functionsBernhard Riemann