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

Lecture

Fourier Series: Periodic Functions

Description

This lecture covers the concept of Fourier series for periodic functions, including the properties of even and odd functions, the calculation of Fourier coefficients, and the representation of functions using sine and cosine terms. The instructor demonstrates how to find the trigonometric Fourier series for specific periodic functions and emphasizes the importance of understanding the properties of functions to determine the coefficients accurately.

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

Maria Colombo

Official source

Related concepts (32)

Sine and cosine

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .

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.

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.

Sine and cosine transforms

In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics. The Fourier sine transform of f(t), sometimes denoted by either or , is If t means time, then ξ is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.

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.

Related lectures (50)

Fourier Series and Equations

Covers Fourier series, periodic functions, and Fourier transforms.

Vibration of Diatomic Molecules

Explores the vibration of diatomic AB molecules and Fourier series, emphasizing equations of motion and Dirichlet's theorem.

Fourier Series: Odd and Even Functions

Explores odd and even functions in Fourier series applications, including half-range series and cosine/sine series.

Integration: Lecture 8

Covers the integration of functions and Fourier series with examples and exercises.

Determinantal Point Processes and Extrapolation