Removable singularity

Par exemple, la (unnormalized) sinc function {style d'affichage {texte{sinc}}(z)={frac {sin z}{z}}} has a singularity at z = 0. This singularity can be removed by defining {style d'affichage {texte{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 {style de texte {frac {péché(z)}{z}}} around the singular point shows that {style d'affichage {texte{sinc}}(z)={frac {1}{z}}la gauche(somme _{k=0}^{infime }{frac {(-1)^{k}z ^{2k+1}}{(2k+1)!}}droit)=somme _{k=0}^{infime }{frac {(-1)^{k}z ^{2k}}{(2k+1)!}}=1-{frac {z ^{2}}{3!}}+{frac {z ^{4}}{5!}}-{frac {z ^{6}}{7!}}+cdots .} Officiellement, si {displaystyle Usubset mathbb {C} } is an open subset of the complex plane {style d'affichage mathbb {C} } , {displaystyle ain U} a point of {style d'affichage U} , et {style d'affichage f:Usetminus {un}flèche droite mathbb {C} } is a holomorphic function, alors {style d'affichage a} is called a removable singularity for {style d'affichage f} if there exists a holomorphic function {style d'affichage g:Urightarrow mathbb {C} } which coincides with {style d'affichage f} sur {displaystyle Usetminus {un}} . We say {style d'affichage f} is holomorphically extendable over {style d'affichage U} if such a {style d'affichage g} existe.

Contenu 1 Riemann's theorem 2 Other kinds of singularities 3 Voir également 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} et {style d'affichage f} a holomorphic function defined on the set {displaystyle Dsetminus {un}} . The following are equivalent: {style d'affichage f} is holomorphically extendable over {style d'affichage a} . {style d'affichage f} is continuously extendable over {style d'affichage a} . There exists a neighborhood of {style d'affichage a} on which {style d'affichage f} est délimité. {style d'affichage 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 {style d'affichage a} is equivalent to it being analytic at {style d'affichage a} (proof), c'est à dire. having a power series representation. Définir {style d'affichage h(z)={commencer{cas}(z-a)^{2}F(z)&zneq a,\0&z=a.end{cas}}} Clairement, h is holomorphic on {displaystyle Dsetminus {un}} , and there exists {style d'affichage h'(un)=lim _{zto a}{frac {(z-a)^{2}F(z)-0}{z-a}}=lim _{zto a}(z-a)F(z)=0} par 4, hence h is holomorphic on D and has a Taylor series about a: {style d'affichage h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+cdots ,.} We have c0 = h(un) = 0 and c1 = h'(un) = 0; Donc {style d'affichage h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+cdots ,.} Ainsi, where z ≠ a, Nous avons: {style d'affichage f(z)={frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+cdots ,.} Cependant, {style d'affichage 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, c'est à dire. 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 {style d'affichage m} tel que {style d'affichage lim _{zrightarrow a}(z-a)^{m+1}F(z)=0} . If so, {style d'affichage a} is called a pole of {style d'affichage f} and the smallest such {style d'affichage m} is the order of {style d'affichage 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 {style d'affichage a} de {style d'affichage f} is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an {style d'affichage f} maps every punctured open neighborhood {displaystyle Usetminus {un}} 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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