Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Unclosed Curves Integrals

In course

MATH-207(d): Analysis IV

The course studies the fundamental concepts of complex analysis and Laplace analysis with a view to their use to solve multidisciplinary scientific engineering problems. ...

Description

This lecture covers the calculation of integrals over unclosed curves, focusing on essential singularities, Laurent series, and residue calculation. The instructor explains the implications of missing points on curves and the closure of integrals. Participants engage in live discussions and examples to deepen their understanding.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

David Valentin Strütt

Sébastien Basterrechea

Official source

Ontological neighbourhood

Mathematics

Analysis: Complex analysis

Related concepts (32)

Essential singularity

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.

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 .

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.

Isolated singularity

In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z0 is an isolated singularity of a function f if there exists an open disk D centered at z0 such that f is holomorphic on D \ {z0}, that is, on the set obtained from D by taking z0 out. Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function is any isolated point of the boundary of the domain .

Singularity theory

In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string.

Related lectures (43)

Holomorphic Functions: Taylor Series Expansion

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Laplace Transform: Analytic Continuation

Covers the Laplace transform, its properties, and the concept of analytic continuation.

Analytic Continuation: Residue Theorem

Covers the concept of analytic continuation and the application of the Residue Theorem to solve for functions.

Differential Forms Integration