Oxford Calculus: Fourier Series Derivation
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- Опубликовано: 1 авг 2024
- 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/tx9dy...
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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: Findi...
Classifying critical points using the method of the discriminant: • Oxford Calculus: Class...
Partial differentiation explained: • Oxford Calculus: Parti...
Second order linear differential equations: • Oxford Mathematics Ope...
Integrating factors explained: • Oxford Calculus: Integ...
Solving simple PDEs: • Oxford Calculus: Solvi...
Jacobians explained: • Oxford Calculus: Jacob...
Separation of variables integration technique explained: • Oxford Calculus: Separ...
Solving homogeneous first order differential equations: • Oxford Calculus: Solvi...
Taylor’s Theorem explained with examples and derivation: • Oxford Calculus: Taylo...
Heat Equation derivation: • Oxford Calculus: Heat ...
Separable Solutions to PDEs: • Oxford Calculus: Separ...
How to solve the Heat Equation: • Oxford Calculus: How t...
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Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
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Finally!!! Please, make a derivation of Laplace Transform!
I appreciate you really did the integrations and didn't just state the results.
I am a computer science student, currently delving around machine learning and it brought me here. I once again am reminded of how beautiful mathematics can be!
Thank you for talking about the interchange of summation and integration and the property of uniform convergence!
Great video!
Adding to the list of people suggesting this: would love to see how to derive the Fourier transform from Fourier series. 😊
Perhaps this is the easiest video on Fourier series on RUclips. Doing integration ( specially showing the value of delta (mn) ) makes this video unique. That's why I easily grasped the concept how to identify the co-effcient. Thanks a lot. My professor just messed up the topic and made it hard to learn.
Studying physics rn, this helped soooooo much! Thx!!!
I loved the derivation. I could only imagine how Fourier felt when he derived or invented this. Seems so clear but we know some steps.
Also reversing the sum with the integral you said we know it converges to f(x). I thought that was what you were trying to prove.
Accompanying FREE worksheet courtesy of Maple Learn here: learn.maplesoft.com/d/DLEJJJNPPUILFLPNCTLRGSJHCTMHCRDJCTAUIRKKGSGPBSFHJNIFNRPODNPFBLJROIHMNIPIOUKLPHNIILISJMFLKULTNLGPHGNS
too good bro. loved it
Please make a video on Fourier, Laplace Transforms and Special Functions such as Bessel functions, Hermite , Legendre Functions.
Really im wating for that
Oh my, just came a minute shy to the end :) Tom, is there a way to describe a FT in terms of SU(2) Lie group? And I didn't watch the video yet - just in case the answer is out there - I'll do in a moment!
I am very interested with!
Just want to say chalk on a board presentation can be very good! Children for a while now never see chalkborads in schools ( here in UK ) and I have been laughed at for being so old as to remember chalk! My dad take about learning to write on a slate?
Hi. Im a 11th grade student. I didnt understand the part where: integrating the sin function, then inputting x=L gives zero (while deriving last RHS term for a0). I get why it happens for a cos function, we integrate it and it becomes sin, and every integer multiple of pi for sin is 0..... but for sin when we integrate, it become cos function... which is not 0 at every integral multiple of pi right? If you could clarify this doubt asap it'd be of help
Each of the 3 positions contains each decimal digit 1/10 of the time (if you count leading zeros). So each digit appears 3×1000/10 = 300 times.
More generally: In a list of all possible n-positional integers (allowing for leading zeros) each base-b digit appears exactly n·b^(n-1) times, e.g. b=10 and n=3 is 3·10^(3-1) = 3·10² = 300.
That seems way easier than how I did it lol. My approach was:
There’s 10 in the first 99 in the ones space (1,…,91) and then 10 more for #s 10-19, giving 20 total in 1-99; this repeats 9 more times for each set of 100 numbers giving 200 total, but then there will be 100 more for every number 100-199 (from the hundreds place) given a grand total of 300😊
For orthogonality, for m=n, you only covered m=n>0, the case m=n=0 has to be treated as a special case or else you are again dividing by 0.
I think he didn't considered it bc he was working with positive integers
Probably pretty low-brow for you, but can you do one on induction proofs?
Why did u decide to integrate this eqation on 23:12 ?
Hi Tom can you make a video testing the new AI chatGPT's maths skills by asking it a bunch of maths questions. I've used it a couple of times and it seems to be quite knowledgeable to a degree.
I had the opposite experience. It came upw ith some incorrect proofs for me but confidently thought they were correct.
The sun(x) is a new one to me! :-)
integration of cos would not be 0 at 30:43, would you please double check when you get a chance please?
It's indded 0, if you plot cos(npix/L) on 2d, you will see that integral is the sum of area over -L to L
Although cos is an odd function, but this integral will be still 0 due to area between x=0 to L/2 is the same of x=2/L to L. One is posivte,the other is negative,so they canceled out. For area between x=0 to -L is the same.
Iam doing
I think the school is good
Does this guy have a youtube tattoo??? Wild.
Is that an english thing, using the tall S as delta instead of the triangle?
it's the small letter delta.
@@MxIraAram Whats that mean? The triangle is only for numbers?
@@beachboardfan9544 Delta is a character from the greek alphabet. Just as in the roman alphabet, they have upper- and lowe-case characters. The lower-case delta is the "S" and the upper-case delta is the triangle.
@@ste1l1 Ahh thanks!
capital Δ lowercase δ
16:00 Odd number times even number is even. Got a bit confused for a while
odd times even is odd
@@evazhang3232 0*1=0. In the integers, even times odd is even. It appears this is not analogous to odd and even functions, but this was just a passing remark that was irrelevant so it doesn't matter that it was wrong.
A derivation of the fast fourier transform pwease
Ouhhh
I feel help me the school please
i guess humanities was never part of your fabulous degrees
your probably right being humanities seems to be not allowed these days
Wow, what kind of a lunatic does this this to his body?
You are extremely rude, what kind of a lunatic you are to leave such rude comment