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

Lecture

Fourier Coefficients of Periodic Functions

In course

MATH-202(c): Analysis III

The course studies the fundamental concepts of vector analysis and Fourier-Laplace analysis with a view to their use in solving multidisciplinary problems in scientific engineering. ...

Description

This lecture delves into computing the Fourier coefficients of periodic functions, focusing on the square wave and the triangle wave. The instructor explains the case distinction for even and odd frequency numbers, simplifying integrals using symmetry properties, and computing the coefficients for cosine and sine modes. The lecture concludes with a discussion on the convergence of Fourier series and Dirichlet's theorem, highlighting the ability to reconstruct the original function for a large class of signals.

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

Instructor

Pablo Antolin Sanchez

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.

Sawtooth wave

The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform. The convention is that a sawtooth wave ramps upward and then sharply drops. In a reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the extreme case of an asymmetric triangle wave.

Square wave

A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions between minimum and maximum are instantaneous. The square wave is a special case of a pulse wave which allows arbitrary durations at minimum and maximum amplitudes. The ratio of the high period to the total period of a pulse wave is called the duty cycle.

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.

Wave

In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a traveling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave.

Related lectures (35)

Fourier Series: Understanding Signals and Transformations

Explores Fourier series, left and right limits, theorem proofs, and exam formats.

Fourier Analysis: Fourier Series and Transform

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

Fourier Series: Harmonic Analysis

Covers the Fourier series representation of periodic signals and harmonic analysis.

Improper Integrals: Convergence and Comparison

Explores improper integrals, convergence criteria, comparison theorems, and solid revolution.

Fourier Series: Understanding Coefficients and Periodicity