# Removable singularity

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

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