how to get the Fourier series coefficients (fourier series engineering mathematics)
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- เผยแพร่เมื่อ 3 ม.ค. 2019
- Learn how to derive the Fourier series coefficients formulas. Remember, a Fourier series is a series representation of a function with sin(nx) and cos(nx) as its building blocks. Meanwhile, a Taylor series is a series representation of a function with x^n as its building blocks. These are two must-know series in your calculus and engineering math classes.
Check out the complex Fourier series here: • Complex Fourier Series...
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Every time I watch one of your videos, my love for mathematics just keeps increasing. Fourier series was never explained like this in any of my classes. We are just told to accept it.
I am glad to hear that you enjoy my videos! thank you!
🤣
Points at Fourier and says it's just a name.
1000000 Engineers feel a cold shiver runing down their spines without knowing why.
LOL exactly
Lol I didn't shiver 😀
Thank you.
Finally I understand what the Fourier is all about.
Comparing it with Taylor is awesome.
You are a great teacher.
I love you, you’re such a great and humble person man. All the best from Italy, watching you to prepare my Calculus Exam!
Elena Clara Maria thank you!!
You're a blessing to calc students everywhere thank you so much
You are so good Mr. Blackpenredpen . Never learned Fourier series in this way. The way you showed the derivation of the formulas made much sense. Thanks a lot.🥺👍
I learned about these recently in my partial differential equations class and I think I can shed some light on why you would multiply by cos(nx) or sin(nx). When Fourier series come up in a PDE the sin and cos terms are eigenfunctions after separating variables. There is a theorem in PDE about Sturm-Liouville differential equations that says that if the DE for the eigenfunctions is in Sturm-Liouville form then the eigenfunctions are orthogonal to each other with a specific weight function, which comes from the form of the DE (and form a complete set). Knowing about orthogonality of functions it would seem only natural to multiply by orthogonal functions. It is just like in linear algebra when you have an orthogonal set you can easily calculate each coefficient in a linear combination using the fact that the dot product is 0 when vectors are orthogonal. One other thing I think is interesting to note is that the coefficient term for a_0 is also the average value of f(x) on the interval. Something interesting and fun to consider is why it would be the average. There are good intuitive reasons...
The term 0 is part of the cosine sum (in reality, the sum goes to 0 to infinite)
@@asxxsss6106 It is, but you cannot get a_0 from plugging 0 in for n in the a_n term. Also, as I said, the expression for a_0 is also the average value of f(x) on the interval from -pi to pi, which you can justify intuitively as well as rigorously.
this was a very good read . very well written indeed friend
Great videos! You're making student lives so much easier. Best math teacher ever! All the best from Bosnia!
please Fourier transform derivation, complex Fourier or other Fourierr-ish stuff
isn't this the fourier transform derivation?
JPK314 no, this is the Fourier series we are dealing with in this video
Finally thank you sooo much!! After being pushed around all over TH-cam and different materials, I actually found something that relates to what I'm doing and it's then broken down and explained gradually in a way our lecturer didn't bother to do. Thank you soo much. I finally understand have a full grasp of what I'm doing. While watching the video, I was also proving the cases so I can defend everything you've taught. Thank you.
I am happy to hear. Thank you!
Great video. You explained it in a really nice (completely perfect I'd say) way and I enjoyed filling in the gaps with the double angle formulas and stuff.
Great presentation.
Have a better understanding of Fourier series now. How this man came up with the series and transform is beyond me.
It's wild to think that he got there because he wanted to solve the heat equation
wow, best work I've seen so far..thanks! wish you can make it longer... I haven't seen any tutorial where they use Taylor series as an analogy, but it helps to have a simpler infinite series as an analogy.
wow ! you are amazing ! 2 weeks lecture of my prof isnt even comparable to your video ! Thanks so much for this video !! helped me A LOT. also your attitude is amazing :) keep going
Glad to hear. Thanks.
Literally studying this right now. Thanks for the derivation!
i like how each time i don't understand a topic and i saw your videos i heave a sigh of relieve......
This happened to be the first time for me to understand this furrier series. Thank you abundantly, sir.
Amazing!!! Beautiful to see where that silly 1/(2*pi) prefactor comes from!!
I figured out that part about m being the same as n! The integrands can be written as (cos((n-m)x)+/-cos((n+m)x)/2, and since we've shown twice that the integral from -pi to pi of cos(kx) is zero when k is nonzero, the only thing that matters is when n-m=0, so we can integrate 1/2 over (-pi, pi).
I know you said you're making a follow-up, I'm just really proud of myself for some reason :)
that was really good explanation of that building blocks. I never knew it came up like that before. thanks for the derivation. now i feel i understand more on the background of fourier series
misugijun thank you!! I use that analogy quiet often. Such as when we solve higher order DE with constant coefficients, e^rt is the building block.
this video actually made me enjoy math! thank you!
Bro you look so happy teaching math, i wish professors and teachers would also have the same passion and love for teaching and solving math problems
Your energy altered my mood too, i was so stressed
Thank you so much for everything
With Love from syria❤
absolutely the best teacher in TH-cam thank you.
This is getting interesting. Carry on with Laplace as well ☺️
Clear and concise! Excellent, thank you!
Love this guy's channel. never stop holding that mic and saving my degree. big love from England
Glad to hear. Thanks.
Respect brother , definitely you explain all the magic for me.
very clear and my curiousity already explained. Thank You Vey Much.. May God bless you
Great teacher, I can now derive Fourier coeffients, thank you so much
Although I have already learned Fourier series, your explanation gave me new inspiration😀
Oh that's really clear, thanks blackpenredpenbluepen! 😊
17:30, “not [a] simp” - a man of culture, I see 😂😂😂
This is so fun. Great presentation!
Thank you so much. You explained it really really well.
Ey, I love you, saludos desde México. ❤
this guy is literally became my math teacher
appreciate your teaching
Thank you, I can now answer a similar question but with different constants for sine and cosine
IT IS SO SATISFYING. THANK YOU!!! I really appreciate you list the first two lines shows the taylor and fourier and proceed everything afterwards. I was always confused that how does the summation of cos and sin come from at the first place. MATH IS BEAUTIFUL! Would you please explain how to get 0 and pi when n not equal and equal to m? orthogonality ? do you have videos on that? Thank you so much!
Well explained 👏 Really helpful
thanks, the vids are addictive
You are a total math badass!! Thank you
bprp,
You introduce cos(mx) and sin(mx) because that is the proper projection onto the (orthogonal) cos (and sin) function spaces, as are geometric projections to figure out values in geometric space, e.g., Euclidean space.
I actually just learned and heard about this from other comments too. Very cool!!! : )
I love you so much thank you for being good at teaching math unlike most professors.
You have no idea, NO IDEA, how hard some of those professors work to do the best they can. Everyone is different and is an individual. Your professor may be saying about you that he wish you were like some other student. Would you appreciate such a comparison of your abilities with someone else's??? I think not. Every professor cannot be like this guy just like you cannot be like Einstein!
I really enjoyed watching 😊verry nice teaching
Can't wait for the next video ^^
: ))))
Great explanation!
Also a 2nd video about non periodic functions would be nice, convergence, when to use Fourier and when Taylor and so on. The Fourier analysis is huge. One of the best parts of Calculus
When we learned the Fourier series in my calc class, it was taught for any interval [-p, p] will you be making a video on this?
Brendan Russell Well, he said you can also do it for 2π instead of π, but considering that the formulas he gave are inner products, it’s implied that it works for any p.
Have you done videos on parametric differentiation and parametric integration?
Explained it quite easily
Your videos are saving my university studies
This is so great ❤
You will make my mechanical engineering career easier man, you are the best!
: )))
@@blackpenredpen what about doing Fourier transform video? :D
I love how you added the note about mx I was starting to stress about it😂😂😂
Well explained thank you very much... One subscription added
After finishing the Fourier analysis start on laurent series
best math teacher ever
Awesome! How about Fourier transform for solving ODE?
Now the complex version with euler's formula 😁
Can you do a video that shows how to go from the Fourier series to the Fourier transform? Thank you so much!
FEELS SO GOOD
Oh at last about FS! Thk you sir!
: )))))
Это очин красивое видео. Спасибо. Really appreciate your hard work
well explained!😊
If cos(nx) is a even function how come on the interval of -pi and pi it equals zero ? 8:53
The term containing bn confused me a lot. Thank you very much sir for explaining it clearly.
Hi love the channel been watching for ages! Just mentioning it seems a bit not-legit swapping between the m-n around for the sum then swapping it back for the final a(subN) constants
Amazing video! I watched three videos since, did not get it and this one finally helped me a lot. They jumped over details i couldn't catch because i get things slow and now this explains everything well. Taylor comparison also came handy for a freshman! Really good work!
Great work
I have 15 years temperature data. I want to remove seasonality variation from the time series data, so how can I get the constant term ao, the amplitude and the phase change by using these given temperature data? thank you
great video!
thanks man, amazing
I am an Indian🙏🙏 and I really like the way of your explanation ,👍👍
it was just....wow.....👌👌
4:20 I don't quite understand the motivation to compare with the taylor series and then decide differentiation vs integration. Why do we need to do differentiate? Why do we need to integrate? What is our end goal with this?
thank you for this good video
Wait, i have not understood when he replaces m with n, help pls! How can he do this?
you should do one on the Fourier Transform as well!! like see where it comes from, maybe similar to 3Blue1Brown’s video but go more in depth! just an idea tho :P
good work!
Thank you! ❤️
Why can you multiply by cos(mx) inside each integral's terms ?
Did you miss out a0/2 for a special reason, or just leave as a0 for initial learning simplicity? Thanks for even seeing this (if you do)
If you take a0/2 instead of a0 , umm.... a0 was a constant and a0/2 is still a constant. So you can take any number, but it needs to be a constant.😊
Thank you so much!!!!!!
amazing . love it
I have seen somewhere DC term is taken as ao and somewhere ao/2. Can you tell please why?
One 20 minute video later and i get what my lecturers spent hours trying to teach us. Very good video happy smiley zero.
Thank you!
thanks man, fantastic work, I was pulling out my hair when my Calculus book didn't explain how to compute a_0
Haha you are the bob ross of maths! Greets from Greece :)
i don't really understand at 14:07, if an is a constant for each term of the summation shouldn't it be like:
a1+a2/2+a3/3+...+an/n= 1/pi integral of f(x)cos(nx) from -pi to pi?
My best lecturer
the best way to think about this and get a true and deep understanding about it, is if your approach it from a linear algebra point of view. the key to the whole thing is inner products and they will solve all yo' problems
BlackpenRedpen, actually, the reason why you should multiply by Cos(mx) or Sin(mx) "f(x)" is because you exploit what's called the "inner product"; I am not 100% sure that this is what Fourier was thinking at the time, but if you know what an innerproduct is, it's simple to see that you have to multiply by a new function to get a specific coefficient. (It's a bit like a dot product except between scalars.. and what do you do when you want to extract the "x" component of a vector? you dot the vector with the unit vector "x", which is sort of what you are doing here basically with cosines and sines)
Great video..
Love frn Nepal.
Can you please do the video showing how the integrals equal π or 0 when m & n are equal or not equal, respectively. I don’t doubt it, I’d like to see it!
Thank you very much
Best explanation 😍😍😍😍
Please make a video on Solution to wave equation in form : £An e ^in2π(x−λt)/L
Could you make a video on how to solve improper integrals like integral from 0 to inf of f(x)cos(x)dx. I can't understand the last part where you use some Jordan identity to demonstrate the arc lenght of the imaginary curve is 0. :(
The TacoBusiness There were no imaginary curves used in this video, what you are talking about is not even relevant to the video because there are no improper integrals in the video. Also, in general, such integrals with a cosine or sine in them diverge, unless f(x) decays sufficiently fast. In those cases, literally just use integration by parts.
Hey
What if we try integration by parts on the A_n formula?
You prob. will have to do that, for example, f(x)=x^2