Oxford Calculus: Fourier Series Derivation

University of Oxford Mathematician Dr Tom Crawford explains how to derive the Fourier Series coefficients for any periodic function. Accompanying FREE worksheet courtesy of Maple Learn here: https://learn.maplesoft.com/doc/tx9dy... Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: https://play.google.com/store/apps/de... Apple: https://apps.apple.com/us/app/maple-c... We start by deriving the orthogonality relations for sine and cosine, which are essential for the derivations of the Fourier Series coefficients. The integral relations rely on the trigonometric ‘product-to-sum formulae’ which enable the product of two sine or cosine terms to be separated and thus integrated directly. The delta function is also introduced to help to simplify the notation. We then assume that a Fourier Series of the required form exists, with as yet unknown coefficients a0, an and bn. These are derived by first integrating the entire equation from -L to L to get a0; then multiplying by cosine and integrating to get the an coefficients for each n; and finally multiplying by sine and integrating to get the bn coefficients for each n. The integrals are evaluated using the previously derived orthogonality relations. Finally, the interchanging of the summation and integral signs is addressed with a very brief discussion of uniform convergence and what this means in the context of a series. Don’t forget to check out the other videos in the ‘Oxford Calculus’ series – all links below. Full playlist: Oxford Calculus Finding critical points for functions of several variables: Oxford Calculus: Finding Critical Points f... Classifying critical points using the method of the discriminant: Oxford Calculus: Classifying 2D Critical P... Partial differentiation explained: Oxford Calculus: Partial Differentiation E... Second order linear differential equations: Oxford Mathematics Open Day 2021: Differen... Integrating factors explained: Oxford Calculus: Integrating Factors Expla... Solving simple PDEs: Oxford Calculus: Solving Simple PDEs Jacobians explained: Oxford Calculus: Jacobians Explained Separation of variables integration technique explained: Oxford Calculus: Separation of Variables I... Solving homogeneous first order differential equations: Oxford Calculus: Solving Homogeneous First... Taylor’s Theorem explained with examples and derivation: Oxford Calculus: Taylor's Theorem Explaine... Heat Equation derivation: Oxford Calculus: Heat Equation Derivation Separable Solutions to PDEs: Oxford Calculus: Separable Solutions to PDEs How to solve the Heat Equation: Oxford Calculus: How to Solve the Heat Equ... Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: @maplesoft Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-c... For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. Facebook: tomrocksmaths Twitter: tomrocksmaths Instagram: tomrocksmaths Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collec...