# Removable singularity Zum Beispiel, das (unnormalized) sinc function {Anzeigestil {Text{sinc}}(z)={frac {sin z}{z}}} has a singularity at z = 0. This singularity can be removed by defining {Anzeigestil {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 {Sünde(z)}{z}}} around the singular point shows that {Anzeigestil {Text{sinc}}(z)={frac {1}{z}}links(Summe _{k=0}^{unendlich }{frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}Rechts)= Summe _{k=0}^{unendlich }{frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{frac {z^{2}}{3!}}+{frac {z^{4}}{5!}}-{frac {z^{6}}{7!}}+cdots .} Formal, wenn {displaystyle Usubset mathbb {C} } is an open subset of the complex plane {Anzeigestil mathbb {C} } , {displaystyle ain U} a point of {Anzeigestil U} , und {Anzeigestil f:Usetminus {a}rightarrow mathbb {C} } is a holomorphic function, dann {Anzeigestil a} is called a removable singularity for {Anzeigestil f} if there exists a holomorphic function {Anzeigestil g:Urightarrow mathbb {C} } which coincides with {Anzeigestil f} an {displaystyle Usetminus {a}} . We say {Anzeigestil f} is holomorphically extendable over {Anzeigestil U} if such a {Anzeigestil g} existiert.

Inhalt 1 Riemann's theorem 2 Other kinds of singularities 3 Siehe auch 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 {Anzeigestil D} und {Anzeigestil f} a holomorphic function defined on the set {displaystyle Dsetminus {a}} . The following are equivalent: {Anzeigestil f} is holomorphically extendable over {Anzeigestil a} . {Anzeigestil f} is continuously extendable over {Anzeigestil a} . There exists a neighborhood of {Anzeigestil a} on which {Anzeigestil f} ist begrenzt. {Anzeigestil 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 {Anzeigestil a} is equivalent to it being analytic at {Anzeigestil a} (proof), d.h. having a power series representation. Definieren {Anzeigestil h(z)={Start{Fälle}(z-a)^{2}f(z)&zneq a,\0&z=a.end{Fälle}}} Deutlich, h is holomorphic on {displaystyle Dsetminus {a}} , and there exists {Anzeigestil h'(a)=lim _{zto a}{frac {(z-a)^{2}f(z)-0}{z-a}}=lim _{zto a}(z-a)f(z)=0} durch 4, hence h is holomorphic on D and has a Taylor series about a: {Anzeigestil 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; deshalb {Anzeigestil h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+cdots ,.} Somit, where z ≠ a, wir haben: {Anzeigestil f(z)={frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+cdots ,.} Jedoch, {Anzeigestil 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, d.h. 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 {Anzeigestil m} so dass {Anzeigestil lim _{zrightarrow a}(z-a)^{m+1}f(z)=0} . If so, {Anzeigestil a} is called a pole of {Anzeigestil f} and the smallest such {Anzeigestil m} is the order of {Anzeigestil 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 {Anzeigestil a} von {Anzeigestil f} is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an {Anzeigestil 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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