Fourier Series of e^x from -pi to pi (fourier series engineering mathematics)

[@thedoublehelix5661]

Pro tip: if you notice a constant is being used often, give it a name. I took (e^pi - e^(-pi))/pi to be equal to gamma and that cleaned up my work a lot! Also at the end you can notice that gamma = 2sinh(pi)/pi.

[@natew7336]

You have been helping me understand math since calculus and I'm in PDE's now. You're a true hero

[@alexander51413]

It's interesting to see that if you ignore the sin(nx) term in the expansion, the resulting function is no longer e^x but now cosh(x) (and ignoring a_0 and cos(nx) gives sinh(x)). I guess it makes sense knowing that e^x = cosh x + sinh x and that probably also explains why lots of comments point out that expansion of e^x can be expressed very easily using the hyperbolic trig functions.

[@crismal6477]

That’s awesome, I didn’t even know that. I’m just learning about Fourier series and it’s really cool, I wish I learned about all this earlier.

[@CharlesPanigeo]

This is not a coincidence! In fact, this is a valid definition of sinh(x) and cosh(x). Every function can be uniquely decomposed as the sum of an even function and an odd function. For e^x that decomposition is e^x = cosh(x) + sinh(x) as you said. Relating that the Fourier series, the cos(x) terms of the Fourier series always describe the even part of the decomposition, and the sin(x) terms always describe the odd part. So we can use the Fourier series to calculate the even-odd decomposition of any function!

[@stumbling]

You have no idea how much I needed this. Thanks. May Chen Lu be with you.

[@The1RandomFool]

Just a minor nitpick: (e^(pi) - e^(-pi))/2 can be abbreviated sinh(pi).

[@resident5124]

This was really cool, I enjoy anything Fourier so seeing this was great. Thanks!

[@tomvanmoer8202]

This was on my Calculus II exam back in 2011. Those were some good times.

[@blackpenredpen]

Yay!!!

[@jasonryan2545]

This is indeed beautiful. For one, I could never understand why some equations look comparatively more beautiful than others. I can see now that if the correct inputs are in place, the equation does the work for us! Once again though, this is indeed a fantastic series! Thank you do much! ❤🎉

[@angelmendez-rivera351]

YAY you made the yay video Also, notice that the term A0 is equal to sinh(π)/π. Just thought it was interesting. In fact, you can entirely factor out a multiple of sinh(π)/π from the series.

[@blackpenredpen]

Yup, it is! : )

[@aashsyed1277]

yaya yay yay yay series aka fouryay yay ya y ay yatyb series!!!!!!!!!

[@brightkwofie982]

Great Sir, i never understand fourier series till i watched this video, Bravoo

[@radiotv624]

Will you be doing Fourier Transform?

[@no1bot_in_Dt]

This is so nice. Proud to see a Nigerian youth be so creative ❤ I wish you success in all you are about to embark on.

[@varshiniarunkumar5643]

Thank you sir....This sum was asked in my internals exam....Now i got this sum clear

[@kevinzenith]

Hands down. Me too! 😂😅

[@stephomn]

that was an amazing result looking forward to more awesome results thanks for sharing and keep it up

[@Àlify.11]

Hello sir I'm from India This was extremely good... Thank you

[@highwastaken]

You can bypass the headache giving integration by just converting everything into the complex form, so computing Cn and then from that computing An, saves you a lot of time if you're familiar with complex numbers

[@NicolasSchmidMusic]

it scared me when those children started yaying

[@horsehorse1791]

Bro you are great yaar Just keep working I appreciate your work 👍👍

[@ozzyfromspace]

Math for its own sake is a beautiful thing, gotta say ❤️

[@chimaru8943]

you're a life saver BPRP #yay

[@blackpenredpen]

You're welcome!

[@Rajgorabhijeet]

Thankyou sir, B. Engineering Learning from India .

[@jokinnn]

thanks, nice trick on the integral

[@krishnak.dileep4120]

Thank you so much.... You are like God to me

[@ok-gl7eq]

Thanks helped a lot

[@siggea5912]

Thanks for the video!

[@nancysharma9454]

I like the way you teach .....

[@Patapom3]

Amazing!

[@shivadahiya1914]

Thanks bro u make my day

[@abdallahallahham8586]

This is amazing ... thank you

[@zabrakhan5178]

How would you visualize this when looking at the graphs of both e^x and the Fourier series of e^x?

[@angelmendez-rivera351]

Zabra Khan The Fourier series takes that segment and the repeats it periodically, resulting in discontinuities at every multiple of π. It repeats it at the same height.

[@ffggddss]

@@angelmendez-rivera351 Right, except the discontinuities are at every odd multiple of π. Fred

[@angelmendez-rivera351]

ffggddss Correct, that is what I meant to say.

[@jeffreyluciana8711]

Excellent!

[@camilomuianga7865]

Why do we have to integrate a none periodic function in the interval [-pi, pi]?

[@mehdizangiabadi-iw6tn]

A=(4,4) B=(-3,-3) C=(A^B) beautiful

[@kamosevoyan4370]

Ты великолепный человек, спасибо большое!

[@NeonArtzMotionDesigns]

I ended up getting a 4 on the AP calculus ab

[@angelmendez-rivera351]

Also, I know you said the Fourier series does not converged at the endpoints. But I did notice that if you let x = π, then the sin(n•x) all become 0, and cos(nπ) = (-1)^n, and if you multiply that by the coefficient (-1)^n/(1 + n^2), then the result is 1/(1 + n^2). So this would give you the sum, assuming the series were to converge to the function at the endpoints. But the problem is that it also works for x = -π,, and that seems to give a contradiction since the series would stay exactly the same, but the thing it equals would be different. It is a pity.

[@blackpenredpen]

Angel Mendez-Rivera Yes, I actually said the exact thing when I recorded the video but decided to cut that off to shorten the length. The interesting thing is, sum of 1/(1+n^2) is equal to the average when we plug Pi and -Pi in. Crazy and I don't know why!

[@angelmendez-rivera351]

blackpenredpen ooh I used to know th explanation but I forgot it. It had something to with this one phenomenon that caused the oscillation to go crazy near the endpoints

[@danielauto3767]

No, not a pitty. The end points do converge. They just don't converge to the the values of the function at either end. Both ends converge instead to the average of the function values calculated at the end points.

[@danielauto3767]

It's called Gibbs phenomenon. Look it up

[@angelmendez-rivera351]

Daniel Auto Yeah, I actually realized that before you commented, and I posted in my long response to the comment in he asked the questions

[@buzeaalexandru4769]

Your videos about calculus are really cool man. I learned a lot from them. Can you do the limit as x approaches 0+ of x^(e^-1/x) in your next video?

[@angelmendez-rivera351]

Buzea Alexandru L = lim (x -> 0+) x^(e^-1/x). log L = log [lim (x -> 0+) x^(e^-1/x)] = lim (x -> 0+) e^(-1/x)•log x, since log x is a continuous function on the interval (0, ♾). e^(-1/x)•log x = log x/[e^(1/x)], so log L = lim (x -> 0+) log x/[e^(1/x)] = lim (x -> 0+) (1/x)/[e^(1/x)•(-1/x^2)] = - lim (x -> 0+) x/[e^(1/x)] = 0, since the form is 0+/e^+Infinity = 0/infinity = 0. Hence log L = 0, so L = 1, and since L = lim (x -> 0+) x^(e^(-1/x)), this means lim (x -> 0+) x^[e^(-1/x)] = 1

[@luisgamalielbasurtocavazos6076]

Thanks you just saved my life lol

[@zedzad3279]

Dear Sir , Can you explain how you get the integration of a0 and a n to the power n ??? please

[@purim_sakamoto]

お疲れ様でした Yes! Say! Fouriyay!

[@i_am_anxious02]

200K (almost) subs #YAY Among the last of the good creators at the heart of this platform #YayYayYayYay :)))

[@blackpenredpen]

Pi is the best 😁😁😁😁😁thank you!!!

[@i_am_anxious02]

blackpenredpen 😁😁

[@victorpaesplinio2865]

I did the same thing, but for the complex version of the series. I got Sum from -infinity to infinity of sinh(pi)/pi (-1)^n e^(i*n*t) All the steps to find the complex coefficient are consistent, but Wolfram Alpha and others sums calculators says this sum diverges. Is this result consistent? I know this videos is more than 2 years old, but I didn't find the complex version of e^x series anywhere on internet.

[@sahanravindu7828]

thank you sir

[@DC-zi6se]

Step jumps are there

[@الملكهالمغروره-ك4غ]

Thank you

[@sugarfrosted2005]

I get it, you're assuming that it has a form then extracting the form using linear operators.

[@angelmendez-rivera351]

sugarfrosted He is assuming it can be expressed as a linear combination of the basis space of waves.

[@ffggddss]

I concur that this deserves 4 yay's! Fred

[@shin81able]

nice video :) what happens if its e^(x-pi) how to calculate this?

[@rshawty]

azy j’ai lâché un like juste pour le début chu un gamin mdrr yaaaaaay

[@shazullahyusufzai5704]

Could you please solve this equation x^x=e

[@zakichahboun543]

hello brother i have a question Can I replace e ^ (π) -e ^ (- π) with sinh (π) ?????

[@apoorvvyas52]

How to deal if the interval is between say any two numbers a and b rather than - pi to pi?

[@AndDiracisHisProphet]

it gets ugly

[@blackpenredpen]

Definitely.

[@maxwellsequation4887]

Intense

[@sciencebyoliver6888]

is this on a open interval from -pi to pi????

[@shivimish9962]

How do you test anything for convergence?

[@emperorpingusmathchannel5365]

Intuitive proof for taylor and fourier series?

[@angelmendez-rivera351]

GLaDOS What exactly do you mean by proving them? They are definitions.

[@nathanisbored]

@@angelmendez-rivera351 they are not definitions, and he just proved them in a couple previous videos

[@angelmendez-rivera351]

nathanisbored He did not prove them. He explained what they are, why we want them, and how to get them.

[@nathanisbored]

@@angelmendez-rivera351 if they were just definitions, then we couldnt use them to derive other things about the functions they represent. for example we wouldnt be able to say the identity he showed at the end of the video is true. we can however define them to be extensions of the domain of a function (for example e^(ix) can be evaluated using the taylor series as a definition)

[@angelmendez-rivera351]

nathanisbored Your comment makes absolute nonsense, considering that everything in mathematics can only be derived from axioms and definitions. The definition of the Taylor series of a function f(x) is the power series with respect to x - a with the coefficient sequence [f^(n)](a)/n!. That has nothing to do with whether there is information about the function we can derive from this series or not. The series already uses information about the function inevitably. The reason we define e^x by its Taylor series is because the function is analytic and the series converges uniformly everywhere, so it occupies the same real domain as the previous definition, but with much more convenience of usage, and this analicity gives a continuation which permits for using this for complex numbers and elements of other topological spaces. That once again has nothing to do with these series being definitions and not theorems.

[@miotegui]

Could be possible that you made a mistake on the a0?. Shouldn't it be a0 = (e^pi - e^-pi)/pi ?

[@Bayerwaldler]

There are different definition for a0. The one bprp uses contains the factor 1/2.

[@miotegui]

@@Bayerwaldler Thanks!

[@azharshah1397]

Sir, can't we use the hyperbolic functions in this problem?

[@thakkarsonal8585]

Kar sake hai

[@kartiksharma7166]

Pls do a video on the summation of x+ x^2 +x^4 + x^8+x^16 ... (inf) Yayayayayayayayayayayayayaya......👍👍👍👍👌👌👌👌this video deserves 💐💐💐💐

[@owamaraayomide8597]

Why is cos(-nπ)=(-1)^n and cos(nπ)=(-1)^n

[@soumyaranjanbehera5295]

is cosnπ is equal to cos(-nπ) ,

[@osuNoobCast]

13:15 what else would i like? i

[@sanzao7487]

Dude, ty!! when i was practicing, i thought that -pi in Sen(kx)-->(Senk-pi) it could not use the identity :c Viva!

[@anntphone7727]

How get 1 power n

[@Jacob-uy8ox]

Tetration of yays!

[@nctbeducation3042]

Thanku bro

[@ccuuttww]

the problem is why we need to approximate the e?

[@blackpenredpen]

I actually have been wanting to do that sum of (-1)^n/(1+n^2) so that's why I do these.

[@VikasSingh-tw8wu]

But the formula for fourier serious also includes 1/2 a0 ...which you didn't write

[@Bayerwaldler]

There are different definition for a0. The one bprp uses contains the factor 1/2.

[@mafujmolla5069]

Great💚

[@_DD_15]

Lol I was gonna say why didn't you use sinh (pi), then I saw that your goal was something totally differe than showing the terms of e^(x) in terms of cos and sin.

[@thomasblackwell9507]

COOL!

[@jinijophy2974]

How did (1+n)^2 become 1+n^2?

[@ffggddss]

It didn't. 1/(1 + n²) came from the factor of 1/(a² + b²) in the integrals, ∫eªˣ sin(bx) dx and ∫eªˣ cos(bx) dx; with a=1, b=n. Fred

[@yossefbudagov8748]

Could you make a video proving the 1/pi approxiamation?

[@rameshparamanikofwestbenga1742]

Right a0/2

[@Bayerwaldler]

There are different definition for a0. The one bprp uses contains the factor 1/2.

[@adarshkumar3184]

Hello ,did you the exact integral of (root over tan(X) +root over cot(X))dx . Please help me guy i'm in trouble

[@angelmendez-rivera351]

adarsh kumar What is that expression supposed to mean? More specifically, what exactly is “root over tan(x)”?

[@adarshkumar3184]

No friend , it is integral of (√(tan(x))+√(cot(x)) dx which is to be determined

[@angelmendez-rivera351]

adarsh kumar He already has a video on these. More specifically, he made a video on integrating the SqRt[tan(x)]. That answers half your question. As for SqRt[cot(x)], notice that cot(x) = tan(π/2 - x). As such, all you must do is perform the substitution u = π/2 - x, and your integrand will simply become a constant multiple of SqRt[tan(u)], and that answers the second part to your question.

[@adarshkumar3184]

@@angelmendez-rivera351 thanks a lot

[@adarshkumar3184]

@@angelmendez-rivera351 I've one more question that is integral of 1over (sin^4(x) +sin^2(X) cos^2(X) +cos^4(x)) whole multiple dx

[@kutuboxbayzan5967]

Well i understanded proof but 1 think is not true e^-pi=lim x->-pi (f(x))=sum An*cos(nx)+bnsin(nx)=-sum An e^pi= lim x->pi(f(x))=sum Ancos(nx)+Bnsin(nx)=-sum An And this meaning e^-pi=e^pi then pi=0. Can you say my wrong?

[@jyotirmoyroy7041]

[@daniellazar4742]

Can you integrate (ln(1/x)+sqrt(x))/(x^2+sin(1/x^2)) ?

[@angelmendez-rivera351]

I am fairly certain this is not integrante using elementary functions.

[@johnk3841]

I like foyay

[@aniketmunde5733]

tq

[@greggregi6383]

you got some big arms

[@gian2kk]

Do you even lift bro

[@JamalAhmadMalik]

#yay I am a wannabe mathfreak!

[@JamalAhmadMalik]

I'm the happiest... You included: #yay #yay #yay

[@blackpenredpen]

Yay!!!!!

[@KaanAkkoyunlu-b9u]

🥶

[@copperfield42]

:o this is more cute than the mess I got XD I didn't simplify those cos(nπ) and sin(nπ), but I transformed those exponential into sinh(π)

[@blackpenredpen]

: )))

[@sub2sub900]

wrong