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

Lecture

Trigonometric Integrals: Residues Method

In course

MATH-201: Analysis III

Calcul différentiel et intégral. Eléments d'analyse complexe. ...

Description

This lecture covers the calculation of integrals using the residues method, focusing on rational functions of sine and cosine. The slides discuss singularities, poles, and examples related to trigonometric integrals.

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)

Marco Picasso

Joachim Stubbe

Official source

Ontological neighbourhood

Mathematics

Analysis: Complex analysis

Related concepts (20)

List of trigonometric identities

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified.

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

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 .

Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. Contour integration methods include: direct integration of a complex-valued function along a curve in the complex plane; application of the Cauchy integral formula; and application of the residue theorem.

Related lectures (31)

Laplace Transform: Analytic Continuation

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

Integration Techniques: Change of Variable and Integration by Parts

Explores advanced integration techniques such as change of variable and integration by parts to simplify complex integrals and solve challenging integration problems.

Generalized Integrals: Convergence and Divergence

Explores the convergence and divergence of generalized integrals using comparison methods and variable transformations.

Unclosed Curves Integrals Log in to Mediaspace to watch this video

Covers the calculation of integrals over unclosed curves, focusing on essential singularities and residue calculation.

Complex Analysis: Laurent Series Log in to Mediaspace to watch this video