how to get the Fourier series coefficients (fourier series engineering mathematics) 

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.

Points at Fourier and says it's just a name. 1000000 Engineers feel a cold shiver runing down their spines without knowing why.

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!

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...

You're a blessing to calc students everywhere thank you so much

Great presentation. Have a better understanding of Fourier series now. How this man came up with the series and transform is beyond me.

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 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 :)

i like how each time i don't understand a topic and i saw your videos i heave a sigh of relieve......

Love this guy's channel. never stop holding that mic and saving my degree. big love from England

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.

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

please Fourier transform derivation, complex Fourier or other Fourierr-ish stuff

Finally thank you sooo much!! After being pushed around all over RU-vid 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.

This happened to be the first time for me to understand this furrier series. Thank you abundantly, sir.

Great videos! You're making student lives so much easier. Best math teacher ever! All the best from Bosnia!

17:30, “not [a] simp” - a man of culture, I see 😂😂😂

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!

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

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.

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❤

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.

this guy is literally became my math teacher appreciate your teaching

Clear and concise! Excellent, thank you!

Amazing!!! Beautiful to see where that silly 1/(2*pi) prefactor comes from!!

I watched all your proof videos. They were awesome. I liked them so much I got hooked on proofs. Now I will watch your Fourier series and FFT videos if they exist. Thank u

This is getting interesting. Carry on with Laplace as well ☺️

absolutely the best teacher in RU-vid thank you.

I love how you added the note about mx I was starting to stress about it😂😂😂

Although I have already learned Fourier series, your explanation gave me new inspiration😀

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)

I am an Indian🙏🙏 and I really like the way of your explanation ,👍👍 it was just....wow.....👌👌

very clear and my curiousity already explained. Thank You Vey Much.. May God bless you

this video actually made me enjoy math! thank you!

Literally studying this right now. Thanks for the derivation!

I'll always say it 0 has got to be my favorite number, it just makes things easier

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?

I love you so much thank you for being good at teaching math unlike most professors.

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

Great teacher, I can now derive Fourier coeffients, thank you so much

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!

The reason for the difference between the integral of cos(nx)*cos(mx) and sin(nx)*sin(mx) where n = m and n =/= m happens because because if n = m, cos(nx)*cos(mx) and sin(nx)*sin(mx), due to being equal to cos²(mx) and sin²(mx), respectively, are always positve whereas cos(nx)*cos(mx) and sin(nx)*sin(mx) can have negative values. Alternatively, you can use the trigonometic identity with the product of two angles for n =/= m. The integral you get is: I(x) = 1/2[sin(nx+mx)/(n+m) - sin(nx - mx)/(n - m)] which equals from -π to π to 0 because n and m are natural numbers/positive integers and with c being any integer, sin(c*π) = 0. In fact, you can use the same integral and use n = m, assuming m approaches n because part of the integral results in a 0/0 situation (more specifically, lim n -> m of sin(nx - mx)/(n - m) = x) and I(π) - I(-π) = [π - (-π)]/2 = π. Of course, you can calculate it classicaly with the power reduction for n = m.

best math teacher ever

Your videos are saving my university studies

4yay! Series! Also, I have a number theory challenge for you. Consider the pair of numbers n - k and k + 1, where n is a natural number and k = 0, 1, ..., n - 1. For what values of n are n - k and k + 1 co-prime (relatively prime, they share no common factors) for all k? In other words, for what values of n is gcd(n - k, k + 1) = 1 for all k satisfying the conditions I established? Good luck with this :) Also, I would like to add to the video the fact that if we instead present the series with a0/2 instead of a0, then we get to present the definition of a0 instead as dividing the integral by π, leaving a symmetry between all the coefficients in that they all divide some integral by π. Also, you can form a Fourier series for any arbitrary interval [-L, L]. Sometimes, for very nice functions, one is even able to let lim L -> ♾ and give an infinitely converging series. Also, this has profound implications from linear algebra. What this means is that the set of sines and cosines with all integer frequencies is a functionally complete orthogonal basis for the vector space. It is very nice. This is why we can do this in the first place, and this moreover reflects the form of the coefficients as being inner products. One final note is that I find interesting how each different series helps in the hierarchy of building functions by using infinite series, allowing us to build more and more functions. Arithmetic allows us to build monomials and binomials. Iteration of this helps us build polynomials. Then we can use infinite polynomials to build smooth functions, which means we can build any sinusoidal trigonometric function. Now we have a series on sinusoidal functions from which we can build even more functions than we can from the Taylor series, allowing us to build step functions, indicator functions, and other waves which are not smooth nor continuous. Then we can form series on those new functions we built to build all sorts of pathological functions, yet allowing us to work with them as if they were not pathological. It also helps lay the foundation for Lebesgue integration. One series allows us to build a basis that we can use to form a series that allows us to build a larger space of functions. Etc. A very useful and interesting hierarchy that I consider to be a happy surprise.

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)

Yo! Supp bro. Are these formulas street legal? Can i end up in joint if i use them in public? Thanks!

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

After finishing the Fourier analysis start on laurent series

Ey, I love you, saludos desde México. ❤

blackpenredpen, you say you don't know why Fourier multiplied by cos(mpi) and then sin(mpi). I know why, because he had the mathematical insight to visualize they were the correct functions (because he was a genius). You say go ask Fourier (can't , he's dead (ha ha). But keep up the good work, you're an inspiration.

One 20 minute video later and i get what my lecturers spent hours trying to teach us. Very good video happy smiley zero.

You will make my mechanical engineering career easier man, you are the best!

I really enjoyed watching 😊verry nice teaching

Explained it quite easily

Well explained 👏 Really helpful

Respect brother , definitely you explain all the magic for me.

Hey BPRP. Will you elaborate on the fourier transform and perhaps its relation to quantum mechanics? I'm specifically talking about Heisenberg's uncertainty principle that you could use as an example of such a transform. Keep up the good work!

Oh that's really clear, thanks blackpenredpenbluepen! 😊

Thanks from India 😊🇮🇳

Now the complex version with euler's formula 😁

I would prefer to use: f(x) = sum_{n=-inf...inf} a_n e^(inx) #EulersFormula makes it much more concise and, moreover, can be used to _derive_ the other expression with sin/cos from this one and vice versa, thus showing their equivalence. (NB: there might be some weird, subtle real-analytic arguments around absolute vs. conditional convergence in some cases that may or may not cause a problem in at least "bad" cases, but I'm not sure about this.) That said, if you want to use real sine/cosine only, I'd also prefer f(x) = sum_{n=0...inf} a_n sin(nx + phi_n) which more closely illustrates that the point of the series is you are summing waves with both different amplitudes a_n and different _phases_ phi_n. Again, equivalence of these expressions is similarly proveable (namely just note that the sin/cos series effectively just takes "half", so to speak, of the phase factors as tau/4 , or 90 degrees.). Moreover, it transparently lets you see that a square and triangular wave are pretty much the same thing, just depending on where you put the phase. One may also be tempted to explore the ominous world of "f"(x) "=" sum_{n=0...inf} sin(nx) subtleetiees concealing #DREAD awwwaaiiitttt..... w000000000000hhhhhhhh.......................................................................

🙂 + 0 + bm pi So sad the second 0 is not as happy as the first

Thank you so much. You explained it really really well.

My best lecturer

Hello. I don't know if you'll answer this but I would like to know if it is ao or ao/2. I have been confused about that. I don't know the correct one. Please help

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

The term containing bn confused me a lot. Thank you very much sir for explaining it clearly.

God bless you From Egypt

This is so fun. Great presentation!

I subbed and went to my second video. The number went from 379k to 380k subscribers, I feel proud!

thanks man, fantastic work, I was pulling out my hair when my Calculus book didn't explain how to compute a_0

Please post more Laplace and Fourier series stuff

Can you do a video that shows how to go from the Fourier series to the Fourier transform? Thank you so much!

Does EVERY function converge in the interval (-pi,pi) using the Fourier Series? And what happens if you change the interval? For example (-3pi,3pi), will it converge too?

Thank you sir form India 🇮🇳

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!

Oh at last about FS! Thk you sir!

thanks, the vids are addictive

Do a video on Laurent series.

You are a total math badass!! Thank you

This is so great ❤

Well explained thank you very much... One subscription added

Haha you are the bob ross of maths! Greets from Greece :)

Thank you, I can now answer a similar question but with different constants for sine and cosine

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

Great explanation!

Please do one on application of Fourier series

King of Fourier.

Can't wait for the next video ^^

Have you done videos on parametric differentiation and parametric integration?

Prof I really love huh......as how as I am I have been struggling with this Fourier but thank Goodness 😅 I came across this I am going to be proving this in my exam in two days time........but there is a big problem I try to confirm what u said if n=m with real figure in the integration I got zero instead of pie that u said.....please I need a quick response thanks 🙏 I am cravingly in love with ur teaching

It's really easy to calculate the integral bounded by -π and π of cos(nx)*cos(mx) dx ,where n is different by m and also solving the same integral but for the function sin(nx)*sin(mx) Yah ,both of them are 0 ;)))

Shouldn`t that be yay^4 ?

Is there a Nobel for teaching? Man if I win a Nobel prize in the future, I am giving you some % of the money I get.

Bro i thought fourier series to be boring in 11th standard but now I am getting loved with it

Great video.. Love frn Nepal.

thank you