In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non-isolated singularities too; those do not have a residue.
Consider an open subset of the complex plane . Let be an element of , and a holomorphic function. The point is called an essential singularity of the function if the singularity is neither a pole nor a removable singularity.
For example, the function has an essential singularity at .
Let be a complex number, assume that is not defined at but is analytic in some region of the complex plane, and that every open neighbourhood of has non-empty intersection with .
If both and exist, then is a removable singularity of both and .
If exists but does not exist (in fact ), then is a zero of and a pole of .
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of .
If neither nor exists, then is an essential singularity of both and .
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i.e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of .
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i.e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at .
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem.
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
thumb|Représentation de la fonction avec deux pôles d'ordre 1, en z = et z = -. En analyse complexe, un pôle d'une fonction holomorphe est un certain type de singularité isolée qui se comporte comme la singularité en z = 0 de la fonction , où n est un entier naturel non nul. Une fonction holomorphe n'ayant que des singularités isolées qui sont des pôles est appelée une fonction méromorphe. Soient U un ouvert du plan complexe C, a un élément de U et une fonction holomorphe.
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non-isolated singularities too; those do not have a residue.
En analyse complexe, le résidu est un nombre complexe qui décrit le comportement de l'intégrale curviligne d'une fonction holomorphe aux alentours d'une singularité. Les résidus se calculent assez facilement et, une fois connus, permettent de calculer des intégrales curvilignes plus compliquées grâce au théorème des résidus. Le terme résidu vient de Cauchy dans ses Exercices de mathématiques publié en 1826. Soit un ouvert de , un ensemble dans D de points isolés et une fonction holomorphe.
Le cours étudie les concepts fondamentaux de l'analyse complexe et de l'analyse de Laplace en vue de leur utilisation
pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.
The course is an introduction to symmetry analysis in fluid mechanics. The student will learn how to find similarity and travelling-wave solutions to partial differential equations used in fluid and c
The course focuses on the current investigations in the fields of fatigue and fracture of composite materials and composite structural components, like adhesively-bonded joints. Students would be able