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

Lecture

Dirichlet Theorem: Complex Fourier Coefficients

In course

MATH-203(b): Analysis III

Le cours étudie les concepts fondamentaux de l'analyse vectorielle et l'analyse de Fourier en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique. ...

Description

This lecture covers the Dirichlet theorem, focusing on the analysis of complex Fourier coefficients. The slides present the theory behind Dirichlet's theorem, explaining the calculation of real and complex Fourier coefficients. The lecture also delves into periodic functions and provides examples of applying the theorem. The instructor demonstrates how to calculate Fourier coefficients for periodic functions, emphasizing the importance of understanding the properties of complex numbers in Fourier analysis.

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

Maria Colombo

Official source

Related concepts (32)

Fourier series

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.

Fourier analysis

In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies.

Joseph Fourier

Jean-Baptiste Joseph Fourier (ˈfʊrieɪ,_-iər; fuʁje; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's law of conduction are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect.

Related lectures (40)

Fourier Analysis: Fourier Series and Transform

Explores Fourier series, DFT, FFT, and their applications in signal processing and spectroscopy.

Fourier Series: Understanding Coefficients and Periodicity

Explores Fourier series coefficients, periodicity, and series relations.

Fourier Analysis: Applications and Inversion Formula

Explores Fourier analysis applications and the inversion formula in the context of periodicity and continuous functions.

Fourier Series and Analysis

Covers Fourier series, analysis, the heat equation, Gibbs phenomenon, and Fourier transform properties.

Wave Propagation: General Solutions