Uploaded December 2022 | Updated September 2026, 2 weeks ago
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: learn.maplesoft.com/doc/tx9dyjwx8o/trm-fourier-series-worksheet
Check your working using the Maple Calculator App – available for free on Google Play and the App Store.
Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en
Apple: apps.apple.com/us/app/maple-companion/id1466659419
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: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S
Finding critical points for functions of several variables: youtu.be/Leomuu82-u8
Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM
Partial differentiation explained: youtu.be/RVwcBGzQcT8
Second order linear differential equations: youtu.be/F54yhRB9qDI
Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E
Solving simple PDEs: youtu.be/uztjxrGY6Jw
Jacobians explained: youtube.com/watch?v=YqMelRryG8U
Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ
Solving homogeneous first order differential equations: youtu.be/uqvqjbAcbL8
Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5g
Heat Equation derivation: youtu.be/rz3cdzZXQms
Separable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwI
How to solve the Heat Equation: youtu.be/l6spigOZCOs
Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
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: learn.maplesoft.com/doc/tx9dyjwx8o/trm-fourier-series-worksheet
Check your working using the Maple Calculator App – available for free on Google Play and the App Store.
Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en
Apple: apps.apple.com/us/app/maple-companion/id1466659419
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: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S
Finding critical points for functions of several variables: youtu.be/Leomuu82-u8
Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM
Partial differentiation explained: youtu.be/RVwcBGzQcT8
Second order linear differential equations: youtu.be/F54yhRB9qDI
Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E
Solving simple PDEs: youtu.be/uztjxrGY6Jw
Jacobians explained: youtube.com/watch?v=YqMelRryG8U
Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ
Solving homogeneous first order differential equations: youtu.be/uqvqjbAcbL8
Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5g
Heat Equation derivation: youtu.be/rz3cdzZXQms
Separable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwI
How to solve the Heat Equation: youtu.be/l6spigOZCOs
Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
