Power Series/Euler's Great Formula
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- Опубліковано 14 вер 2010
- Power Series/Euler's Great Formula
Instructor: Gilbert Strang
ocw.mit.edu/highlights-of-calc...
License: Creative Commons BY-NC-SA
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Awesome, I'm 73 and it's a real joy to do mathematics like this!
Volker Block Awesome, I'm -6 and it's a real joy to do mathematics like this!
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 :)
you must be a real lonely person!
Nate Davis so you are not even born yet? o.O
I'm 24i and I really enjoy this!
If you want mathematics equivalent to Beethoven's symphony or Picasso art, watch professor Gilbert Strang's lectures. This man is a true genius.
The 3 are great, but incomparable imo
Beauty
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
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.
Those arm movements. Gotta love Gilbert.
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.
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.
I just love you Professor Gilbert Strang.....You are the best Professor without any doubt
Gilbert you have done it, yet again just like in the old days.
That demonstration of the Euler formula the derivation of e^theta.x = cos theta + i.sin theta was beautifully done.
I love this guy. Dedication and professionalism.
¡Astonishing! I love this guy.Thanks a lot Professor Gilbert Strang. You are a completely legend.
Great lecture. You make it easy to learn. Thank you for sharing your knowledge with the world
superb as always! Thank you Professor Strang for this wonderful series of lectures..
I was born in 1944 and I am also impresed. What a beatiful exposition
A very satisfying derivation of Euler's famous identity. Superb.
Absolutely Amazing.Learnt something new.Thanks.
Simply the best ! I love him!! Make easy all importants concepts
Mind-blowing! Excellent explanation!
Superb Instructor - really smart ! This is the start of wave functions ...quantum physics.
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.
+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.
Huge Respect! Thank You.
Genius and eloquent educator ..
How hard and sincere in explaining things awesome ❤️❤️❤️.
I like the quality of this video .. KEEP GOING
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.
This makes so much sense!
i love his lectures.
Fantastic teacher... good stuff
Sir thats amazing you explained every bit of it in a very beautiful and clean way thank you so much ❤
So, is there an audience behind the camera, of is he giving us the Dora treatment?
+Sammy Bourgeois some people may call it pedagogy
I feel like his classes only have a handful of students. The man is very talented, but sometimes hard to follow.
JFJSKHDKFDSK I'VE NEEDED THIS FOR A LONG TIME IT EXPLAINS SO MUCH THANKSS A LOT MIT
Gilbert Stang...you are a rock star
Thank you Mr.Strang
fantastic for those who want to clear their concepts....
Beautiful explanation. Well done
This is the most beautiful mathematics I can even conceive of. :' -)
Fantastic presentation.
Great lecture, thank you.
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.
I thought physics is easy to understand than mathematics, but when you teach mathematics it is easiest than anything. Thank you Sir.
Wow...superb...Thank you very much sir...
Thank you for your awesome lecture
Amazing explanation
Wonderful explanation.
amazing lecture
Thanks for making it so clear
Great explanation!!
P.S. you get change the speed to 1.25 or 1.5 if you’re in a hurry!
Tab aur Nahi samjh mein ayenga..
You can use Google translator.
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
Very nice teaching method from India.
Gilbert Strang is the best
dat boi euler inadvertently proving pi as being transcendental
I don't think that the transcendence of pi is proved by Euler...
now, fight!
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.
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.
Super and awesome about your teaching
thats nice i also have another version of deriving Eulers formula of complex numbers!
so for the geometric series, x can't be 1 but it can be bigger ? and it can be smaller ?
Excellent teaching
Excellent lecture 🙏🙏🙏🙏🙏
Seeing me watching this lecture must the equivalent to watch a deaf person sitting by the radio enjoying a good music.
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.
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?
OMG! Great!
Excellent.
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.
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.
i love the sound of writing
at around 30:00 , that series adds up to 2 for x=1 right? great lecture btw
+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)
+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)
Only if you consider n as an even number
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.
is it an art class?
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!!
Euler's formula for series accelerates their convergence
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 ?
yeah me too. But he's really good at explaining though
Superb
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?
1+2+3+... Is a divergent series.
at 9:26 when he says x to the fifth is Strang talking about the fifth derivative of the function f(x)?
+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)
I didn't quite understand the last example. shouldn't the LHS be negative infinity and the RHS infinity?
No, LHS is -(-INF), thus INF. Limit of ln(x) as x approaches 0 is -INF.
I'm a bit confused, isn't this called "Maclaurin Series"?
AFAIK Taylor Series is a more general expansion, not dealing with x = 0
Maclaurin series are just special cases of Taylor series in the same way that squares are just special cases of rectangles.
mrahmanac yes
A maclaurin series is a taylor series where a = 0, otherwise where the function is at x=0
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?
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?
Awesome, I'm 14 and it's a real joy to do mathematics like this!
+Joshua Watt good luck
+Joshua Watt Try out Number Theory! Highly addicting stuff! :D
I'm 12
@@alexandermizzi1095 I'm 1
Sir amezing
I was calling Oiler, Uler till now.
I like pronouncing it Uler better, but it's wrong :(
lol he is german..... why not say his name how he says it?
cory6002 Euler was swiss! (spoke german)
On what principle do we assume a function can be approximated as a power series?
+Mohammed Safiuddin on certain intervals of convergence the limit does not go on to infinity, therefore we can assume the series can approximate some function, plus some error R. if the limit of R approaches zero, then we can assume that after enough iterations of the summation that the series will equal the function exactly.
This course is aimed at high school level or people outside mathematics. He simply skipped proving cos and sin are analytic.
@userisdosser This was uploaded quite recently
Dr. Strang is mathematics' answer to James Stewart.
so did mine :D
The best art in math is infinity. But i'd rather hear it when this Professor say infinity, "it's going forever".
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
Fun video
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.
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.
Holy shit I finally know why e^pi*i = -1 now. This is an incredible day.
This gives further clue on Ramanuhan number summing up as 1+2+3+4 converges to _ஶ்ரீ
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.
I'm 18 and I really need this for tests...Lmao
Is there any PART2
Visit MIT OpenCourseWare at: ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/ to see the full list of topics covered by Prof. Gilbert Strang. Best wishes on your studies!
I don´t know how, but every video is more surprising than the previous one!!! I´ve understood imaginary numbers.
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)
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!!!!"
Plan: aeroplane/series
I'm 97 I love solving hard integrals