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. Ajude a melhorar este artigo adicionando citações a fontes confiáveis. O material sem fonte pode ser contestado e removido. Encontrar fontes: "Removable singularity" – notícias · jornais · livros · acadêmico · JSTOR (Julho 2021) (Saiba como e quando remover esta mensagem de modelo) A graph of a parabola with a removable singularity at x = 2 Em análise complexa, 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.

Por exemplo, a (unnormalized) sinc function {estilo de exibição {texto{sinc}}(z)={fratura {sin z}{z}}} has a singularity at z = 0. This singularity can be removed by defining {estilo de exibição {texto{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 {estilo de texto {fratura {pecado(z)}{z}}} around the singular point shows that {estilo de exibição {texto{sinc}}(z)={fratura {1}{z}}deixei(soma _{k=0}^{infty }{fratura {(-1)^{k}z^{2k+1}}{(2k+1)!}}certo)=soma _{k=0}^{infty }{fratura {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{fratura {z^{2}}{3!}}+{fratura {z^{4}}{5!}}-{fratura {z^{6}}{7!}}+cdots .} Formalmente, E se {displaystyle Usubset mathbb {C} } is an open subset of the complex plane {estilo de exibição mathbb {C} } , {displaystyle ain U} a point of {estilo de exibição U} , e {estilo de exibição f:Usetminus {uma}rightarrow mathbb {C} } is a holomorphic function, então {estilo de exibição a} is called a removable singularity for {estilo de exibição f} if there exists a holomorphic function {estilo de exibição g:Urightarrow mathbb {C} } which coincides with {estilo de exibição f} sobre {displaystyle Usetminus {uma}} . We say {estilo de exibição f} is holomorphically extendable over {estilo de exibição U} if such a {estilo de exibição g} existe.

Conteúdo 1 Riemann's theorem 2 Other kinds of singularities 3 Veja também 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 {estilo de exibição D} e {estilo de exibição f} a holomorphic function defined on the set {displaystyle Dsetminus {uma}} . The following are equivalent: {estilo de exibição f} is holomorphically extendable over {estilo de exibição a} . {estilo de exibição f} is continuously extendable over {estilo de exibição a} . There exists a neighborhood of {estilo de exibição a} on which {estilo de exibição f} é limitado. {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 {estilo de exibição a} is equivalent to it being analytic at {estilo de exibição a} (proof), ou seja. having a power series representation. Definir {estilo de exibição h(z)={começar{casos}(z-a)^{2}f(z)&zneq a,\0&z=a.end{casos}}} Claramente, h is holomorphic on {displaystyle Dsetminus {uma}} , and there exists {estilo de exibição h'(uma)=lim_{zto a}{fratura {(z-a)^{2}f(z)-0}{z-a}}=lim_{zto a}(z-a)f(z)=0} por 4, hence h is holomorphic on D and has a Taylor series about a: {estilo de exibição h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+cdots ,.} We have c0 = h(uma) = 0 and c1 = h'(uma) = 0; Portanto {estilo de exibição h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+cdots ,.} Por isso, where z ≠ a, temos: {estilo de exibição f(z)={fratura {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+cdots ,.} No entanto, {estilo de exibição 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, ou seja. 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 {estilo de exibição m} de tal modo que {displaystyle lim _{zrightarrow a}(z-a)^{m+1}f(z)=0} . If so, {estilo de exibição a} is called a pole of {estilo de exibição f} and the smallest such {estilo de exibição m} is the order of {estilo de exibição 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 {estilo de exibição a} do {estilo de exibição f} is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an {estilo de exibição f} maps every punctured open neighborhood {displaystyle Usetminus {uma}} 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