Power Series/Euler's Great Formula

Power Series/Euler's Great Formula
Instructor: Gilbert Strang
ocw.mit.edu/hig...
License: Creative Commons BY-NC-SA
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[@volkerblock]

Awesome, I'm 73 and it's a real joy to do mathematics like this!

[@n8with8s]

Volker Block Awesome, I'm -6 and it's a real joy to do mathematics like this!

[@emilmeme1717]

Ha! Awesome, I am not a 73, nor a 6, and it is real joy to do mathematics like this, too! The power of X :)

[@qtip3998]

you must be a real lonely person!

[@Kashados]

Nate Davis so you are not even born yet? o.O

[@TripedalTroductions]

I'm 24i and I really enjoy this!

[@skoolwal3874]

If you want mathematics equivalent to Beethoven's symphony or Picasso art, watch professor Gilbert Strang's lectures. This man is a true genius.

[@pianoforte17xx48]

The 3 are great, but incomparable imo

[@wisdomokoro8898]

Beauty

[@yuradew]

This formula is magic and beautiful. My mind was blown when I had first seen this formula articulated then proven with such ease and elegance, not to mention seeing its practical applications in harmonic motion, differential equations

[@rsassine]

I wish my Calculus prof back in my college days introduced the Taylor Series like Prof Strand did. What a great, great teacher. Viva Gilbert Strand.

[@juggerlaplata]

Those arm movements. Gotta love Gilbert.

[@cesarjom]

Wonderful lectures... Dr Gilbert Strang is meticulous in the way he stitches the ideas together to build a wonderfully clear picture of some mathematical topic.

[@georgesadler7830]

Sine and cosine waves are the backbone to the communication system on this earth. This lecture shows how important these two functions are in our daily lives. DR. Strang really lays out Euler's great formula from top to bottom.

[@sammybourgeois5072]

So, is there an audience behind the camera, of is he giving us the Dora treatment?

[@juggerlaplata]

+Sammy Bourgeois some people may call it pedagogy

[@putinscat1208]

I feel like his classes only have a handful of students. The man is very talented, but sometimes hard to follow.

[@fernandodominguez1]

I was born in 1944 and I am also impresed. What a beatiful exposition

[@sngash]

Great lecture. You make it easy to learn. Thank you for sharing your knowledge with the world

[@marienbad2]

That demonstration of the Euler formula the derivation of e^theta.x = cos theta + i.sin theta was beautifully done.

[@salvatorecardamone7717]

+Mohammed Safiuddin If a function is analytic, it can be expressed as a power series, by definition. This is a fundamental concept within mathematical analysis.

[@bgdx.5049]

I love this guy. Dedication and professionalism.

[@moisessoto5061]

Gilbert you have done it, yet again just like in the old days.

[@shahzaibmalik9948]

I just love you Professor Gilbert Strang.....You are the best Professor without any doubt

[@emylrmm]

A very satisfying derivation of Euler's famous identity. Superb.

[@nandakumarcheiro]

The lecturers discussions has inspired me to think more on i tpe sine wave wave pulses that oscillate towards imaginary may be able to record more on computer chips.In between 1and zero the x function may be integrarated to give an inverse function from logarithmic function.This may be differentiated from an inverse function towards logarithmic function.This means any blue crystal absortion may be along absorbing imaginary i type pulses as rotation along e^itheta may follow a typical conjecture that moves along imaginary vertical axis as it converges at that particular real axis.

[@volkerblock]

very nicely said, but unfortunately I'm too stupid to understand this answer. Or should I meditate a little longer? By the way, you have a nice long name. It just takes a little while to sign.

[@biggerthaninfinity7604]

Great explanation!! P.S. you get change the speed to 1.25 or 1.5 if you’re in a hurry!

[@GC-tz1lh]

Tab aur Nahi samjh mein ayenga.. You can use Google translator.

[@idealpotatoes]

JFJSKHDKFDSK I'VE NEEDED THIS FOR A LONG TIME IT EXPLAINS SO MUCH THANKSS A LOT MIT

[@vieiralessandra]

Simply the best ! I love him!! Make easy all importants concepts

[@miqueiassteinle2541]

Melhor é ver uma aula de séries de potência em inglês do que assistir uma só série em inglês. A relação trigonométrica com o número imagiário é muito interessante no contexto de série de potência. E os quâdros dessa sala de aula são muito legais seria bom que todos os quâdros de aula tivessem esse mecanismo.

[@WoWitsGeorgii]

dat boi euler inadvertently proving pi as being transcendental

[@kennylau2010]

I don't think that the transcendence of pi is proved by Euler...

[@Simson616]

now, fight!

[@juancarlosserratosperez8462]

¡Astonishing! I love this guy.Thanks a lot Professor Gilbert Strang. You are a completely legend.

[@edwardj3070]

this is nothing less than the foundation of modern technological civilization. all our children should understand this by the time they have finished high school

[@ArhamKhan05]

Sir thats amazing you explained every bit of it in a very beautiful and clean way thank you so much ❤

[@jansvedman3876]

Superb Instructor - really smart ! This is the start of wave functions ...quantum physics.

[@daydreamer05]

I thought physics is easy to understand than mathematics, but when you teach mathematics it is easiest than anything. Thank you Sir.

[@Taterzz]

i remember on a math test i used this way to define e^x. probably one of the most interesting applications of taylor's series i've ever seen.

[@rohitjagdale7352]

Huge Respect! Thank You.

[@dekippiesip]

Have you been studying that in high school? Her in the Netherlands we don't go farther than integral calculus in high school. Just calculating surface areas or solids of revolutions is as far as it get's in high school.

[@DilipKumar-ns2kl]

Fantastic presentation.

[@MuhammadWaseem-gd1yv]

fantastic for those who want to clear their concepts....

[@szyszkienty]

Mind-blowing! Excellent explanation!

[@creamcheese3596]

In the UK this topic is covered in A-Level Further Maths, studied by 17 to 18 year olds. They study it before they even get to university. I think this video highlights how low US university standards are compared to the UK's.

[@freeeagle6074]

America has AP calculus at high school which is equal to 3 credit points of a 1st-year college course. The AP is similar to the A-Level calculus in UK. America has an scientifically-designed education system that starts with easy-to-understand concepts in an area but very quickly goes to in-depth ideas. Seminars arrive at the pinnacle where you read about 4 recent papers every week (sometimes a book) on a topic and each study group usually gives a presentation every week. Many final projects from the seminar are publishable at academic conferences or journals.

[@elamvaluthis7268]

How hard and sincere in explaining things awesome ❤️❤️❤️.

[@walidnouh1747]

Genius and eloquent educator ..

[@nimrod2330]

superb as always! Thank you Professor Strang for this wonderful series of lectures..

[@andrewcottrell2278]

I have used complex numbers to solve sinusoidal AC electric circuits for years. Just recently, i had been looking at e^jx and derived from this, Cos x + j Sin x. But I can't ever remember anyone proving to me that the given power series of Sin and Cos, ARE in fact true. and at last, I have been enlightened! (Did Taylor's series decades ago!!!) Oh, IF you're an electrician, you use j, not i!!

[@shohamsen8986]

Gilbert Strang is the best

[@AyushBhattfe]

I was calling Oiler, Uler till now.

[@wedeldylan]

I like pronouncing it Uler better, but it's wrong :(

[@cory6002]

lol he is german..... why not say his name how he says it?

[@scp3178]

cory6002 Euler was swiss! (spoke german)

[@mohammadabdallah1956]

i love the sound of writing

[@rekhanarsapur3125]

Gilbert Stang...you are a rock star

[@MrPabloguida]

Seeing me watching this lecture must the equivalent to watch a deaf person sitting by the radio enjoying a good music.

[@karsunbadminton7180]

Thank you Mr.Strang

[@mplaw77]

Wonderful, wish you had been my Calculus prof. I did well enough but I just memorized, thick book so not much time to actually think.

[@holyshit922]

Euler's formula for series accelerates their convergence

[@onderozenc4470]

Euler formula works for all practical reasons but what is somewhat peculiar is that although the Euler formula is obtained at x = 0 it comes true for all x"s values ?

[@markwheeler202]

Great lecture! I've never seen this done before. That being said, he missed a huge opportunity at ~25:00, where he could have quickly shown one of the most amazing facts in mathematics. What happens when theta = pi? e^i[pi] = cos [pi] +i sin [pi] cos [pi] = -1 sin [pi] = 0 therefore... e^i[pi] = -1 (Apologies for the notation)

[@AaronHollander314]

Fantastic teacher... good stuff

[@pappaflammyboi5799]

Dr. Gilbert: "I have to bring in the imaginary number 'i'. Is that okay? Just imagine a number 'i', ok? And everybody knows what you're supposed to imagine..." Students: Was that supposed to be funny? Why is nobody laughing? Did I miss something? Looks at notes... Classic Gilbert deadpan pun.

[@aymenjerbi1587]

Well, this is not a very "strict" mathematic proof. you cannot tell that d(x-> sum(a(k).x^k))/dx = sum (k.a(k)x^(k-1) unless you verify that you have the right to do so. x->exp(-1/x²) is a counter example.

[@s.kphysicsandmath1o111]

Very nice teaching method from India.

[@tonymaric]

This is the most beautiful mathematics I can even conceive of. :' -)

[@CatsBirds2010]

i love his lectures.

[@MichaelCurrie]

Dr. Strang is mathematics' answer to James Stewart.

[@sailorgaijin8838]

Absolutely Amazing.Learnt something new.Thanks.

[@af8811]

The best art in math is infinity. But i'd rather hear it when this Professor say infinity, "it's going forever".

[@abdelrahmangamalmahdy]

I like the quality of this video .. KEEP GOING

[@burakbey21]

For a Princeton student body, they sure do ask a lot of basic questions and it interrupts the flow of an otherwise great lecture. You can kind of sense the frustration of the instructor at a couple points

[@companymen42]

This is the basis for all of electrical engineering. It pisses me off so much that my circuits instructor on the first day of my first EE class didn't go "remember that one random formula that you learned in calc 2? It is the basis of your ENTIRE FUCKING MAJOR!!!!"

[@SpinWave]

Beautiful explanation. Well done

[@jagareksa.bahureksa]

Plan: aeroplane/series

[@rameshdas]

Superb

[@peon3715]

I don´t know how, but every video is more surprising than the previous one!!! I´ve understood imaginary numbers.

[@kishorekumarbehera6704]

Wow...superb...Thank you very much sir...

[@mrahmanac]

I'm a bit confused, isn't this called "Maclaurin Series"? AFAIK Taylor Series is a more general expansion, not dealing with x = 0

[@SilverArro]

Maclaurin series are just special cases of Taylor series in the same way that squares are just special cases of rectangles.

[@muhammadrafaqat7748]

mrahmanac yes

[@danielmiddleton6094]

A maclaurin series is a taylor series where a = 0, otherwise where the function is at x=0

[@xhourglazzezx]

This makes so much sense!

[@n8with8s]

My calc 2 professor did a similar thing in one of his lectures. I prefer the proof that uses vector calculus, however. It's a lot less convoluted.

[@R00KIEo87]

What should I not correct the no factorial of 1 is not concluded is -1 because you're trying to score points against the individual person which are writing off a mathematical sum because you made a fault

[@venmathikannan1255]

Super and awesome about your teaching

[@elamvaluthis7268]

Wonderful explanation.

[@Shockszzbyyous]

so for the geometric series, x can't be 1 but it can be bigger ? and it can be smaller ?

[@javierarmandodiazcarbajal9846]

Thanks for making it so clear

[@joshuawatt7028]

Awesome, I'm 14 and it's a real joy to do mathematics like this!

[@youmah25]

+Joshua Watt good luck

[@Darkenedbyshadows]

+Joshua Watt Try out Number Theory! Highly addicting stuff! :D

[@alexandermizzi1095]

I'm 12

[@Epic-so3ek]

@@alexandermizzi1095 I'm 1

[@KyujinSim]

Thank you for your awesome lecture

[@JSSTyger]

I'm sorry but I found the presentation to be all over the place and I think he made it very difficult to understand. Example at 6:15 his expression for e^x is wrong.

[@esakkithirugnanam6626]

Excellent teaching

[@nandakumarcheiro]

This gives further clue on Ramanuhan number summing up as 1+2+3+4 converges to _ஶ்ரீ

[@nandakumarcheiro]

This gives further further information on Ramanujan sum 1+2+3+4........converges as -1/12 as it enters a cos x power series the condition under which it becomes a negative value linked with Reimann function that oscillate along real plane of x axis suddenly enters a plane of imaginary axis at-1/2 of Reimann axis.This means Reimann conjecture oscillate along real sine axis suddenly jumps towards imaginary cos function plane giving peculiar information on Reimann conjecture series of a function. Sankaravelayudhan Nandakumar.

[@physjim]

It's interesting that at 26:50 he starts saying that 1+1+... gives infinity, but we know that 1+2+3+...=-1/12. Since we can decompose the sum 1+2+3+...to multiples of 1+1+1+... how can we get one way -1/12 and the other way infinity?

[@theoleblanc9761]

1+2+3+... Is a divergent series.

[@ASSA-dy2ho]

That was maclaurin's series not Taylor's. Am l ri8? Yeah i know maclaurin's series is a special case of Taylor's series

[@ethanmullen4287]

Amazing explanation

[@mattweippert7254]

Holy shit I finally know why e^pi*i = -1 now. This is an incredible day.

[@KIRYUCO69]

can somebody please help me figure out why the imaginary number i cannot be assumed as a constant and become ie^ix when first derivate e^ix?

[@NationalPK]

I'm 97 I love solving hard integrals

[@jons2cool1]

One question I have is if this Euler's identify actually approximates the function. Some Taylor series only approximate values close to the origin, is there a proof that shows this is a valid approximation?

[@jons2cool1]

I'm thinking that for any values you put into the formula that return 'i' the series is divergent. How is that different than a variable that isn't actually a real number, like infinity in the exponent and saying, this impossible number times pie is equal to this. Or even give real value for the impossible number. I feel like 'i' isn't being viewed correctly because you can conceive it in your mind and in certain cases, it returns a real number. But when it is impossible, I don't understand why you are able to assign it a value. Just like 'i' we can have cases of indeterminate sums that converge with L' Hopital's rule. Can you multiply a real number by an imaginary number? 2*infiinty is infinity. I can see how you get '2i' as sqrt(4(-1)), which reduces to 2*i, but can we actually do this? How do we know these same basic mathematical concepts apply when dealing with this imaginary number?

[@surendrakverma555]

Excellent lecture 🙏🙏🙏🙏🙏

[@tenishotshot]

Troll, go away. You are in the presence of greatness. This is like a world-class racecar driver teaching how to use a clutch. This stuff is easy and not even an inkling of what this man is capable of. The most introductory class at MIT for undergraduates. Don't embarrass yourself again.

[@sagarikabhagade682]

I didn't quite understand the last example. shouldn't the LHS be negative infinity and the RHS infinity?

[@berserker8884]

No, LHS is -(-INF), thus INF. Limit of ln(x) as x approaches 0 is -INF.

[@1471emre]

Great lecture, thank you.

[@stephenc6185]

But why did Euler decide to make all the derivatives of the Taylor series to match at x=0? Is that just how mathematicians think?

[@charleslyell3748]

You have to make x=0 to kill all the terms that contain x in order to calculate the coefficients of the series. The first coefficient is a0, then take the first derivative, plug in x=0 and get the term a1, and so on. So, a0 = f(0), a1 = f'(0), a2 = f''(0), and so on.

[@WahranRai]

Why taking pi (3.14...) for computing sin(x) and cos(x) !!! By assumption we are developping around x=zero !!!

[@jupatj24]

Because Pi is a nice number to compute trigonometric functions. It doesn't matter which values you choose to evaluate Euler's formula, the formula will be valid. Again, we choose x=0 to develop the formula because that's the most convinient thing to do.

[@joefagan9335]

WahranRai Good point. He lacks a little rigour here and doesn't show that the Taylor (well Maclauin series) converges everywhere to the function he's trying to represent. It happens to converge everywhere for e^x, Sin and Cos (which blows my mind!) to those functions and so it converges at Pi.

[@energie9966]

thats nice i also have another version of deriving Eulers formula of complex numbers!

[@wolfnederpel]

at around 30:00 , that series adds up to 2 for x=1 right? great lecture btw

[@rosskemptheboss]

+Wolf Nederpel I think the infinite series of 1/n diverges, so the right hand side would 'add up' to infinity (even though it's counterintuitive)

[@rosskemptheboss]

+Wolf Nederpel I think the infinite series of 1/n diverges, so the right hand side would 'add up' to infinity (even though it's a bit counterintuitive)

[@zakariarakhrour9158]

Only if you consider n as an even number

[@KevinAlexandair]

amazing lecture

[@chrisbrown865]

thankyou sir fascinating

[@newton464]

OMG! Great!

[@NirajC72]

at 9:26 when he says x to the fifth is Strang talking about the fifth derivative of the function f(x)?

[@salvatorecardamone7717]

+NirajC72 No, he means x to the fifth power, i.e. x*x*x*x*x. The derivative terms in the Taylor expansion for sin(x) are equal to either 1, 0 or -1. Typically if one wants to denote the derivative of a function, a prime will be used, e.g. df/dx = f'(x), d^2f/dx^2 = f''(x), etc. Alternatively, where the prime becomes cumbersome at higher order, you can use Roman numerals, e.g. d^5f/dx^5 = f^v(x)

[@SigfriedNothung]

Euler the greatest mathematic

[@kenichimori8533]

Power Series Euler's Point to Number is 354 √0

[@kenichimori8533]

P/E

[@v3le]

is it an art class?