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Lecture# Analyse IV: Laurent Series and Singularities

Description

This lecture covers the analysis of Laurent series and singularities, focusing on the properties of meromorphic functions, poles, essential singularities, and regular points. The instructor addresses questions related to convergence radius, holomorphic functions, and the use of residue theorem. Participants engage in discussions about the nature of singularities and the choice of gamma0. The lecture also delves into the calculation of Laplace transforms, inverse Laplace transforms, and the application of Sturm-Liouville problems. Various examples and questions are presented to clarify concepts and deepen understanding.

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In course

MATH-207(d): Analysis IV

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.

Instructor

Related concepts (77)

Display resolution

The display resolution or display modes of a digital television, computer monitor or display device is the number of distinct pixels in each dimension that can be displayed. It can be an ambiguous term especially as the displayed resolution is controlled by different factors in cathode ray tube (CRT) displays, flat-panel displays (including liquid-crystal displays) and projection displays using fixed picture-element (pixel) arrays. It is usually quoted as , with the units in pixels: for example, means the width is 1024 pixels and the height is 768 pixels.

Computer monitor

A computer monitor is an output device that displays information in pictorial or textual form. A discrete monitor comprises a visual display, support electronics, power supply, housing, electrical connectors, and external user controls. The display in modern monitors is typically an LCD with LED backlight, having by the 2010s replaced CCFL backlit LCDs. Before the mid-2000s, most monitors used a CRT. Monitors are connected to the computer via DisplayPort, HDMI, USB-C, DVI, VGA, or other proprietary connectors and signals.

Singularity (mathematics)

In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the function has a singularity at , where the value of the function is not defined, as involving a division by zero. The absolute value function also has a singularity at , since it is not differentiable there. The algebraic curve defined by in the coordinate system has a singularity (called a cusp) at .

DisplayPort

DisplayPort (DP) is a digital display interface developed by a consortium of PC and chip manufacturers and standardized by the Video Electronics Standards Association (VESA). It is primarily used to connect a video source to a display device such as a computer monitor. It can also carry audio, USB, and other forms of data. DisplayPort was designed to replace VGA, FPD-Link, and Digital Visual Interface (DVI). It is backward compatible with other interfaces, such as HDMI and DVI, through the use of either active or passive adapters.

Removable singularity

In complex analysis, 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. For instance, the (unnormalized) sinc function, as defined by has a singularity at z = 0. This singularity can be removed by defining which is the limit of sinc as z tends to 0. The resulting function is holomorphic.

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