How to compute a Fourier series: an example

[@965productions9]

I have been trying to understand this for the past 12 hours and this video from 12 years ago is the biggest help I've found. Thank you so much sir!

[@MadeyeMoody492]

You know what they say. Better 2 hours before your final than never...

[@gergerger53]

After such a frustrating day trying to organise all my printouts and really make a dent in my understanding of this, your video comes along at 1am and now I feel like I can sleep tonight. THANK YOU.

[@RossJenk1]

You are quite literally, single handedly saving my degree. My lecturers notes are shambolic compared to those which you produce, just a shame you're not my Engineering Mathematics II lecturer!

[@frozenrats]

How is life for you now 10 years later?

[@RossJenk1]

@frozenrat pretty good, thank you. Engineering wasn't really for me, I had a better head for business. So that's what I did, set up a medical engineering company and now hire 4 engineers.

[@diags_1109]

@@RossJenk1 That's cool, also interesting looking at a video you watched 10 years ago haha

[@DrChrisTisdell]

Wow - I'm really glad and humbled to receive a comment like yours. If you liked this video then there are lots more examples in my new (free) ebook, where each lesson is linked to a video of mine. The link is in the description. Good luck with Fourier series!

[@bloomingbeautybyjessblesse1519]

The book is no longer available ): do you have another link? please help!

[@arialamoosai3966]

The best lecture i have seen on the Fourier Series ! Thank you .

[@airfusionalok]

Thank you

[@RossJenk1]

Well my university have decided to screw me over because I had to get an operation and long story short I am now sitting this exam. Again. This video for a second time has helped me more than I could have imagined and the PDF is great! The one thing I'd suggest is possibly worked solutions but I appreciate how much time has already gone in to it out of your own goodness. Keep up the good work, much preferred channel over others such as Khan Academy. If I could hire for private tutoring I would!

[@Fatkaido]

Thank you Dr Tisdell. Im having an exam tomorrow and this is what's been keeping me awake until you appeared on my search results. I wish you the best as well🎉

[@martafajlhauer8877]

I had 100% on my ODE mid-term thanks to you, now it's time for my final. I would love to repeat my score :) As my lecturer is not the best I can say that I had my score only thanks to Dr Chris. Thank you.

[@martafajlhauer8877]

Ha! Thanks for the reminder. Do you know that I got 80%+ for my ODE in the end and I have studied for a week or so. I work as Data Scientist contractor at the space industry company. I think that these ODE went pretty well but will revise them one day. Highly recommended!

[@Ensign_Cthulhu]

As always, you're crystal clear and your examples are easy to follow, even for someone who hasn't had formal university maths. I could quite happily watch these vids all day.

[@nablarnermk8844]

Never lose trust in a 14 year old calculus video! Best video on this topic by far

[@DrChrisTisdell]

Thanks for the feedback. Glad you are finding these things of some use. Next year I'll post a full course on PDE with a new, free ebook. Good luck with your studies.

[@mikey071113]

If only there were more professors like you online, I'd have a much greater understanding of all my subjects! Thanks!

[@2020LLL]

finally someone who doesn't take for granted that everyone is born genius. Excellent tutorial and wonderful practice for us starters who have full-time everything going on yet wants to better themselves in understanding step by step and not skipping just because we should already know this... Please, more examples; and as you get more in depth, your followers will become more as well but with a greater concept than those who were born engineers.

[@Arnab1actor]

definately the easiest way on the internet to understand the concept of fourier series.. kudos!!!!!!

[@DrChrisTisdell]

Hi - thanks for the kind words. I have been at UNSW, Sydney for 10 years. During that time I, too, have learnt a lot from my colleagues. I've had a bit of experience with teacher training and get the trainee teachers to "critique" some of my videos, eg "What would you do differently?". You may also like to use YT as a learning tool in your own classroom at some stage. Good luck with teaching!

[@ManuelMendez1]

So many thanks from Dominican Republic, a VERY simple way to teach no simple things

[@DrChrisTisdell]

My pleasure! My current mantra is: free videos + free ebooks = better education. I'm glad I could help.

[@Tofumoode-jy6ro]

wow, you managed to explain what I've spent for over 6 hours trying to yet understand. Many thanks. :D

[@ortek87]

I got a total of 83 from differential equations course and got a BA. In the past, I would be walking on the clouds for such a grade, but now I feel for that missing AA :))) Thank you Dr. Tisdell, blessings to you

[@MDMAx]

On behalf of all the undergraduate viewers I THANK YOU for your videos. Science bless technology advanced 21.st century.

[@allinengineering8168]

I saw your video 5 years ago in my undergraduate degree now I am watching again in masters degree you are wonderful both lessons going well

[@zayed9902]

Wow😮, you upload this video before i start kindergarten and now i watch it for my first year in engineering college

[@RandyLott]

I haven't seen too many Australian people on these sort of videos. There's something about the accent that I love even though I'm American! Great work Chris.

[@davidfof13]

I really appreciate your work Dr Tisdell. Your explanations are very clear and you speak slowly allowing us, the viewers to easily adapt to the rhythm. many thanks. this vid has widened my vision of fourier series

[@GermanMedinaNeria]

Dr , please to clear this issue: How do you get x values on 5:25 and how do you get Cos NPi -1 (I want to clear -1) on 5:48. Thanks in advanced again,

[@troid]

Thank you for doing this! It means a lot for us students and learners. You've just covered two weeks of class in our school (which is sad). I just hope teachers would learn how to teach like you. Some of them feel they knew everything that they don't even want to learn new things (better way to teach).

[@ozzyfromspace]

Bonus for having a ‘Stralian accent 😊🥳🎊🙌🏽 Awesome video Dr. Chris! Greetings from the US

[@luthandondlovu8613]

DAMN DAMN DAMN WOW.......You have simplified the workload for me after trying many videos

[@praptisalaskar6281]

just a minor correction, at 5:27, the integration of sinx would be -cosx and not cosx. Good video :)

[@earl8295]

No, he brought the -1 out of the parantheses already. Notice the -6

[@DrChrisTisdell]

Look in my playlist "Fourier series"

[@chekwubeugwu4015]

Dr please what do I do to know this Fourier series. I have given it so much time but it seems not to sink in my memory

[@colloupdated]

@@chekwubeugwu4015 u have to believe in ur mind and ur ability to learn,as well as practice 💯 ...u will be gud

[@saadamiens]

You are the best teacher on youtube

[@ortek87]

Sir, thank you for this. I think the true instructor should possess the characteristics that you have. Ours are always reluctant to give details about how to calculate, they even skip the intermediate parts in examples, they say it is our duty to find out how the calculations are carried out and our job to refresh our memories from calculus to carry out difficult integration by parts etc. 4.5 hours to my final exam and thank you for making these available to us. You sir, are a great man!

[@BlakLotus]

I had no idea what my book was saying. You just saved my butt. Thanks a bundle!

[@alreadytakenindeed]

A guys like you, Chirs are giving the math a good name. You gave me hope :)

[@jackdavy9182]

This really helped me to understand when you can integrate from [0,L] instead of [-L,L]

[@ragnarlothbrok4090]

Sir you are best lectirer you deserve more recognition!

[@sai1120]

My life would be a lot easier if you are my maths lecturer, Thanks a thousand!!!

[@srijandhakal3277]

Watching from Nepal sir🇳🇵 ,really helpful for bachelor degree.

[@cemcalsar7367]

thats what ı have been searching for, so you made it simple ,hocam thanx to you so much

[@DrChrisTisdell]

I am not sure what you are exactly asking but if you mean you want to sum over all natural numbers (and not just odd numbers) then you would replace "n" with "2k-1" everywhere and in the summation sign you would sum from k=1 to infinity.

[@drdread]

Thanks a lot for the brilliantly laid out and thorough example. Saved me a lot of headache! 2nd Year Mechanical Engineering student at University of Nottingham

[@DrChrisTisdell]

Hi Steve D'Alembert's solution is used to solve PDEs with ONLY initial condtions. Separation of variables is used when there are both initial and boundary conditions. It's important to understand which method to use when. With D'Alembert's solution, you assume that the solution is of the form $u(x,t) = \phi(x + ct) + \psi(x - ct)$ and then you want to determine the functions $\phi$ and $\psi$ from the given initial conditions. Hope this helps!

[@DrChrisTisdell]

@ronalddlelariarte I doubled the integral, but halved the length of the interval of integration. The reason is that it is nice to have 0 in the integral sign sometimes because is simplies calculations. It comes back to thinking of the area. Instead of the areas cancelling (as is the cases with a_0 and a_n), with this b_n the area is repeated (doubled). Hope this helps.

[@anannyauberoi7984]

Great explanation! Love the little simplification in the end- thanks a ton!

[@lifematch]

can you make more video of Fourier series. i really need to grasp this concept and your video was the clearest among other vids in terms of what to do with Fourier series. i would very much appreciated as will other students around the world. nonetheless thank you soo much for the lesson.

[@DrChrisTisdell]

Hi. Please have a look at the annotation that gives you a link the a whoel video about odd and even functions.

[@madgary187]

mate you make fourier series look soo simple unlike any of the maths professors at university of edinburgh.

[@Beriszle]

Hello, I am an aerospace engineering student getting ready to start my second year and I am just curious about this sort of thing. Just wanted to say very clear explanation and VERY interesting!

[@PokemonParadise2010]

Dr. Tisdell You have done a wonderful job with this video. This is excellent. I learn a lot from your videos. Please make more videos. You are great. Regards, SO

[@Jonorocks999]

Top Video, really clear. Now Fourier is sorted for my exams...

[@khaaaled2007]

Thanks :) I'll watch all of them tonight , exams are just around the corner thanks again

[@mexico442]

Wonderful job Dr. Tisdell. I should let you know that I've passed calculus because of your videos. Thanks!

[@confidencedavid8890]

Most simplified..I've seen so far🙌

[@DrChrisTisdell]

@jackofnowhere Did you watch the video linked at 2:27? Maybe the best (and simplest) way to understand odd functions is to rotate the graph of the function 180 degrees about the origin and to see if you get the same graph as the original function.

[@06Wanderer]

sir u are the best i couldnt understand what is this bt after ur video i can finally clear my exam

[@Andiotic]

Thank you very much sir, this is one of the most straight forward, easy to understand example I've seen so far.

[@studiowomi7723]

we are greatful for provisions of your reference.from proff.Wanyoyi and Abbas from Kenya.

[@rcomid]

Thanks a lot Dr. Tisdell for this video and please accept my warm regards from Iran.

[@DrChrisTisdell]

Hi - both forms are acceptable (and are equivalent). In my formula I divide by 2 in the formula for a_0, so that the FS is: a_0 + (other terms). It is equivalent to define a_0 without dividing by 2, so that the FS will be: a_0/2 + (other terms).

[@gremlinextreme101]

it can be quite easily proved that any continuous function can be written in as a unique sum of an odd and even function. An even function can be written in the form (f(x)+f(-x))/2 and an odd function can be written in the form (f(x)-f(-x))/2, you can use these definitions to show general cases of odd*even or odd*odd and to show that a differentiable odd function has an even derivative.

[@sisip123]

I'm currently cramming for MATH3121 .. this was good revision. thanks!

[@MigLMariano]

Thank you so much, this was a great video to help refresh my memory for my final exam

[@animalmother9848]

At 5:17, Can anyone please tell me how or why, 6 pi is divided? Would very much appreciate it thanks!

[@Lownayto]

Thanks for uploading this, can't understand my actual professor. I love you forever good sir.

[@DrChrisTisdell]

Hi Fartx: the $f(x)$ is usually a given function (given initial temperature). If $f$ is not specified then you can't do anything more than just write down the integral form for the $b_k$ as you have done above. Good luck!

[@RaviSharma91]

@HaBaBaMBiZ in fourier series these are following conventions even X even = even odd X odd = even odd X even = ODD do not mix it with simple maths multiplications. good luck

[@resmond]

@mayabentz Do you mean odd or even - Odd is 180 degree symmetry about the x axis; even is symmetrical about the x axis If you meant real or imaginary - A real function (not sure if you'd use that term) has no imaginary parts (being = j = sqrt (-1) ).

[@DrChrisTisdell]

Thanks! Hope you find the free ebook of some use also!

[@ew0054]

Thank you so much for this video! You have taught me in 10 minutes what I have not been able to learn in class for the past month. I will be sure and pass this along to my friend who is equally struggling. I would like to ask, at 4:35, when you double the integral and 1/2 the length, what property allows this?

[@mcardlel90]

I wish you were taking my lectures! I can actually understand what you are saying... Thank you

[@parveensharma75]

Amazing explanation Chris Keep up your good work.

[@ASFAnarchist]

Great! I remember learning this a few years back, but I urgently needed a refresher. This was exactly what I needed. Thank you!!

[@shivanshusiyanwal296]

I have a quiz on fourier series in an hour , I was panicking earlier but now I am not. Thanks. Nice accent.

[@DrChrisTisdell]

That's it DMan - you got it!! Just in case you forget, if you follow my "is my integrand even or odd?" method (in the vid) then you can't go wrong!

[@thegreatgiginthesky]

Thank you - you just helped me to pass my University Maths for Physics exam! The clearest example I have seen.

[@kingrobin12]

Man this is AWESOME!! You saved my exam!

[@DumboSanchez]

Youre a legend Tisdell love your teaching style!

[@SethyTan]

Thanks! you're better than my school lecturer.

[@WaqasKhan411]

Thanks Dr. Chris for uploading such an informative video

[@ecdmuppet]

Dr. Tisdell, Thanks for this video. Could you post a more complex example with perhaps a sawtooth or triangle wave? The integrals get hairy when f(x) is more than just a constant.

[@niftyusername]

Very clear and awesome. Understanding is such a stress relief!

[@bearded-cat]

thanks a lot! I wish we had such lecturers in my uni

[@chasemarangu]

it was great to see a practical application of the formulas instead of more circle gifs EDIT: I want to make the circle gifs myself, I just didn't know how to find the aᵢ and bᵢ

[@pedrosa3142]

@11phax Two products can be integrated using the method of parts. I am not sure how to visually describe this but it can be found as a definite integral.

[@DrChrisTisdell]

The value of L is always half the period of the function. What is the period of f in this example?

[@sanchezdel8]

Wow! This is the explanation I've been looking for. VERY CLEAR. THANK YOU!!

[@fartx211]

Hey Chris, you know how in the end of the Heat Equation questions, when trying to figure out the that last constant Bk by using the condition that u(x,0)=f(x), resulting in a Fourier sine series. We use those formulae that were in the Fourier series topic. But this requires us to to integrate f(x)sin(n*Pi*x/L). How can we do that if we dont know what f(x) is? Or is it there somewhere and i cant see it? I mean, for some reason, the working suggests f(x) is even, but how do we know that?

[@23rdnova]

Dear Dr Chris i would appreciate is you could make a video explaining how to compute a complex function

[@DxBoi]

hey chris just a question, even though i think it's a bit of a stupid one. how did u get [cos(n*pi) - 1] to equal [(-1)^n - 1] u said something about oscillating...what if it wasn't an oscillating sequence? what would you write then?

[@tsehayenegash8394]

I have 15 years temperature data. I want to remove seasonality variation from the time series data, so how can I get the constan term ao, the amplitude and the phase change by using these given temperature data? thank you

[@jirehroxas9307]

do you have a video that shows how to solve for An when it is an even function?

[@DrChrisTisdell]

Hi Torpy, separation of variables is too long to cram into an sub-10-minute vid, but I am experimentng on doing this another way......stay tuned!

[@DxBoi]

thanks heaps Dr Chris! i see now what u were talking about...and thanks so much for all these awesome vids! i feel they're the best way to get started on a topic. cheers!

[@fartx211]

try drawing a simple y=cos(x) graph. notice how at 0, 2pi, 4pi, 6pi.... y =1, and at pi, 3pi, 5pi, y=-1 so at every odd number multiplied by pi, cos would equal -1, and at every even number multiplied by pi, cos would be 1. you could therefore write cos(n*pi) as (-1)^n, since when n=0,2,4,6...., (-1)^n = 1 and when n=1,3,5,7...., (-1)^n = -1, so you can replace cos(n*pi) with (-1)^n

[@convolucion1981]

@BarGesq what if you aren't given that the period is two? how do you it's two? --Suppose, even, that the domain with which you are given in the piecewise is not the typical (-L, L). What do you do if it's like 0,2*pi or other or another???

[@claudiawellman9854]

How do you know if the function is even or odd? Is there a video I can watch on it?

[@aluminous0]

Thank you so much! I was smashing my head against the wall for hours trying to figure this out before.

[@GermanMedinaNeria]

Please, On 1:50 why the funcion is odd ? Thanks in advanced again !

[@nazlcanpulat4264]

Thanks for your tutorial.It is so clear! :) But only thing is, I didn't unterstand at 5.40 seconds , how you get 2 in front of the integral?

[@nicodemusmd]

It's an even function with limits of integration from -1 to 1 so he changed the limits to 0 to 1 and multiplied by 2.

[@fahyiensher8220]

Michael Nicodemus when ever I change the limit then I should multiple by 2?

[@bhagswag8434]

Let me explain for future students. An even function is like having a mirror image across the y axis, right? So lets say my original picture was a simple box; y = 3. A box with height 3. Let's say the width is L = 1, so the box has dimensions of 3 (height) x 1(width). Now, imagine that across the y-axis is a reflection of the box. Another box with same size. Now, if you put both boxes together, you have a bigger box, still with height 3, but length is now 2, total. What's the area under both boxes total? That would just be Integral of (3dx) with limits -1 to 1. OR, because with know ONE box is exactly half of both boxes combined, we can just take ONE box integral and multiply it by 2. integral of (3dx) with limits from -1 to 1, and then multiplied by 2. Same principle applies to cosine. It is EVEN, making a perfect reflection on the other side. Sine is odd, it makes and UPSIDE DOWN reflection, which is when the total area from -L to L cancels.