Ramanujan Integral

[@blackpenredpen]

Whoa!!! (To be honest, I haven’t watched the entire video. I just woke up not long ago. But it would have been a whoa anyway)

[@shivimish9962]

Can you please make a video on the same integral in a simpler manner? I'm not able to get the higher level of maths here

[@Sitanshu_Chaudhary]

@@shivimish9962 bro first to understand milenium transform

[@rrr1304]

Fan

[@anish3084]

Understanding ramanujans mathematics is not your cup of tea blackpen

[@MaksymCzech]

Bruh, this is some hardcore math. I've taken complex analysis class and I think we solved something similar by going from real numbers to complex plane. Can't be 100% sure though. Thanks for your video!

[@googleuser3481]

Proud to belong to S. Ramanujan's country. Love ur contents sir. BTW a fun fact, Ramanujan figured out most of his mathematical formulae & identities in his dreams(his "belief" was some Hindu Goddess provided him with all these stuffs). He used to sleep with a pen & paper beside him and developed a habit of 'sleep writing'. And, most of his discovery papers are now lost which had some of the best mathematical stuffs, whose proofs & derivations are still to be found.

[@jake_runs_the_world]

Google User 🙄🙄

[@sherlockjunior8612]

Jake believe it or not, it's true! Ramanujan actually did dream of these formulae and others called it "intuition"... Because Ramanujan wrote complex formulas etc. Without any proof. He just wrote them down and since he didn't "derive" them, people disliked him.

[@Rishi-he7hs]

@@sherlockjunior8612 No!!! Sir I am also an Indian and please recognize his intuition.... Then why did not Devi appear in dreams of priests or why did not Jesus or Allah appear in any other people's dreams??

[@Swybryd-Nation]

Great application Dr. Peyam Yes Ramanujans MASTER FORMULA. He used it quite often and prof. Moll et al and I wrote a paper about it. Berndt gives a formal proof using finite difference operator E.

[@Jaeghead]

At around 7:35, how do you know that this is the correct choice for phi(N)? If you had chosen a different third root for 1-i you would have gotten a different answer in the end, right?

[@imgayasheck595]

"Srinivasa Ramanujan credits his mathematical findings to the Goddess of Namagiri. According to Ramanujan, she appeared to him in visions, proposing mathematical formulas that he would then have to verify." I believe that he had dreams of proofs, I have had the same where my brain thinks of beautiful mathematical equations when allowed to freely wander such as in a dream

[@darkseid856]

Exactly . Sometimes when I do hardcore study , formulas and equations starts to appear in my head very randomly . The only thing is that I am still stupid at maths.

[@emanuelmartinez3585]

I really love when calculus and number theory meet! 🥰

[@guyguy1811]

This integal never gets old

[@Handelsbilanzdefizit]

When I was younger, ramanujan woke my interest in continued fractions :-)

[@evangelosnikitopoulos9623]

The fact stated at the beginning of the video (the one on which this whole calculation is based) is totally false as stated because the sequence (φ(n))_n does not uniquely determine φ on the rest of the real line (or complex plane), no matter how nice φ is. What are the actual assumptions on f and/or φ? Or rather, what is the actual statement of this Ramanujan Theorem that is cited?

[@theoleblanc9761]

Totally what I think! I wrote a comment the video roughly like you and Peyam don't answer me...

[@sergioh5515]

I've been away from mathematics for a while due to other classes so I found this really refreshing and AMAZING!!!!!!!!! Thank you Dr Peyam for always presenting awesome results :)

[@atrath]

I am a simple math guy. I see Dr Peyam, I hit like! :D

[@andygregory2390]

Excellent presentation. The Indian Clerk lives!!

[@drpeyam]

Andy!!! Hope you’re doing well :)

[@MooImABunny]

Ramanujan's master theorem has some hidden assumptions that you did not take into account. To demonstrate, consider this: take a function phi(x) which generates f(x) by the definition given. Now let phi2(x) = phi(x) + sin(pi*x). Obviously, at integer x, the values are the same. Since f needs the integer only, the function generated from phi2 is identical to f(x) According to the formula, you can reconstruct phi by integrating f(x)x^(s-1). Since the f is the same, the integral is the same, which means the reconstructed phi"s are the same. That's not good, remember, they differ by sin(pi*x). Same goes for any difference that vanishes at integer values of s. We can even ignore the negative integers!! As you can see, that's an issue with reconstructions. When you try to reconstruct a function from limited information, you must know what are the conditions and restrictions on the function, or else your reconstructions can only be approximately true. (This is very similar to Nyquist stuff)

[@theoleblanc9761]

I don't understand the theorem at all...in the hypothesis of the theorem (incomplete to me: for which s..? conditions on f to ensure the integral converges..?), there is Φ (in the power series) but Φ is only define (known) for integers greater or equal than 0, so in the conclusion, if -s is not an integer greater or equal than 0, what is the good value for Φ(-s) ? Indeed, even if there is a good explicit formula for Φ(n) (eg n^2+1), nothing, really nothing states that Φ(z)=z^2+1 for all complex z. More, if there is no nice formula, maybe Φ is some "random/chaotic" function (eg Φ(n)=p_n, the nth prime number) then, what is Φ(-1/3) ?

[@theoleblanc9761]

@Arsene1412 yes but to evaluate the integral we need to understand the theorem to apply it. By the way there no "Φ converge for all s", Φ is a function not an integral or a series and to evaluate the integral we need to have compute Φ on inputs (-1/3) where Φ is not initially defined in the hypothesis, that is why it makes no sense (to me). Why on earth Φ(-1/3) should be the value given in the video, it seems arbitrary. Indeed, you are just given f, so Φ(0), Φ(1), Φ(2),... are naturally introduced by uniqueness of power series but Φ(-1/3) is completely off-topic and relates to nothing with f, our only start point. In fact for me it seems more "moral" to define Φ on every complex input by the by the formula (a way of extending Φ).

[@robertmeadows4852]

I’ve been learning about contour integration and complex integrals but I can’t get my head around Analytic Continuation, Riemann Surface and Conformal Mappings. There aren’t many helpful videos, especially on Riemann Surfaces. Wondering if anyone can help?

[@CHRISTIANZEGARRA2018]

There are great videos about those topics. I took complex analysis last semester

[@HilbertXVI]

I'd recommend getting a good textbook on complex analysis instead, videos only get you so far

[@siddharthpandya7763]

@@HilbertXVI videoed are distracting as well , very few info and takes too much time

[@CHRISTIANZEGARRA2018]

@@HilbertXVI actually videos sometimes help the reader visualize the material in a different way. When I read the Laurent series I was confused and when I found a video it made it more clearly and definitely made me more conformable

[@ricbisa6092]

What a nice result!!!! Keep doing awesome integrals

[@awez_mehtab]

Got an ad in which jimin was singing "life goes on".. 💜 First time I saw a whole ad of 3 minutes

[@robertmeadows4852]

The second he said Mellin Transform I panicked but it seems you don’t need to know about it to understand the video. Really cool way of solving such a crazy integral.

[@immasoxfanbaby]

Thanks my daughter is taking calculus and she needs help. I'm gonna share this with her to learn solutions of calculus

[@googleuser3481]

Ramanujan's Biopic, "THE MAN WHO KNEW INFINITY" gives us some glances of some of his similar mathematical discoveries.

[@drpeyam]

Love that movie!!!

[@Handelsbilanzdefizit]

Yesterday I figured out, when you write: ln(Gamma(x)) = ln(x-1) + ln(x-2) + ln(x-3) + ... You can derive both sides, and you'll get a simple sum for the gamma derivative. But I still don't know, what's the inverse of the gammafunction :-(

[@ArsentyKambalin]

Thnx for an advance content! Could you make much more advanced video? (about method of steepest descent or something else)

[@shahinjahanlu2199]

Hi . I only see your old videos in KZbin. Where is your new videos?

[@drpeyam]

Check my channel and click on videos

[@michelkhoury1470]

Nice solution ! It can also be computed by using differential equations or complex analysis or Laplace transform...

[@soumyadipsarkar5078]

And another thing,phi(N) is not that chunk,it is except that N! 8:54

[@mathranger3586]

Can you integrate (x^2+3)/x^6(x^2+1)

[@عمرانآلعمران-و7خ]

Thanks a lot Dr Peyam ,it is amazing integral .Could you verify the Mellin transform given in this video.

[@ivanlazaro7444]

I think he used the "Ramanujan's Master Theorem". U can find the prove in some books

[@silasrodrigues1446]

I just want to see a closed form for the solutions from this kind of problems. I know that they can't be expressed in terms of elementary functions... But would you leave me dream alone?

[@holyshit922]

I think that Laplace transform will be faster

[@federicopagano6590]

Notice when we arrive that integral= 1/3 Im(junk) we could introduce inside the factor e^(-sx) apply the laplace transform and then finally take the limit as s goes to zero we get exactlty the same result (i tried it wothout watching the final result and then checked it was the same!! I feel like ramanjuan-oid lol

[@akkumar8768]

Proud to be an INDIAN🇮🇳🇮🇳🇮🇳🇮🇳😍😍😍😍😍😍😍

[@Anonymous-lw4nq]

Don't be so proud, no Scientists or mathematicians like to be recognized by their country first.

[@akkumar8768]

@@Anonymous-lw4nq make your concepts clear...u don't know the INDIANS ...Country comes first for every Indian. And your comment literally makes no sense at all.

[@Anonymous-lw4nq]

@@akkumar8768 Don't be over nationalist, it's good to be so, but "too much" can become harmful for their own deeds, example? Germany (pre WW2), and I'm Indian btw, and I hate over patriotism, country is just a made up thing, imagination, but an individual is self made. You aren't the reason for Ramanujan's genius mind nor anyone else except his parents, he was born poor and that time no one helped him except one of his clerk friend in Madras and his mentors GH Hardy and Littlewood in Trinity, Oxford. So don't take fake pride.

[@akkumar8768]

@@Anonymous-lw4nq PROUD TO BE INDIAN. I WILL SAY 10000 OF TIMES... did i tell u that i taught him????And i know about him so dont say here. I WAS SAYING THAT HE WAS AN INDIAN AND I M TOOOOO THATS IT ..HE MADE OUR COUNTRY PROUD ..dont know y are U so much jealous of comment.... LAST LINE FOR U======DONT BE ANTI NATIONALIST!!!!!!.....

[@Anonymous-lw4nq]

@@akkumar8768 Lol Anti Nationalist? I just said don't be over nationalist, taking pride is good, but take pride for the man first then the country, after all whole field of mathematics is proud of him, not just India. You don't need to show everywhere, it itself gets visible.

[@sanjaycosmos9679]

plz sir make more vedios on ramanujuan what he invented in mathematics... i am very fasonited on ramanujuan mathematics.. i am totaly inspired of his mathematics

[@ahmadkalaoun3473]

That's amazing :-) And please in next videos change the position of tge camera because what's on the right side of the board isn't very clear... And thank you ;-)

[@drpeyam]

It’s an old video, the angle has been fixed

[@andygregory2390]

It's fine in HD

[@shanwil474]

I don’t think I’m here yet, I’ll be back in the future when I learn more

[@sadface7457]

Sorry to be late. This problem looked similar enough to fourier transform. That you might be able solve it if you introduction j \omega then take the limit as omega approaches -j (so it becomes one), you can split the exponential to get the right power and use the euler form of the trig function. So I tried.

[@researchersworld4718]

"Master Theorem" it's proof is based on "mathematical analysis" not as "Applied Proof ". It's used to prove differential equations whose solution are in the form of infinite series of algebraic terms.

[@soumyadipsarkar5078]

you missed one thing,that angle is -pi/4 not -pi/2 7:06

[@reetanshukumar1865]

I have better proof, just using gamma function on the complex plane and answer matches with yours but as youtube has no option for the photo as a comment, I can't upload here. but I have a different question to asked, how to integrate (x)e^ix dx. if I used Ramanujan theorem, the outcome is ridiculous

[@Galileosays]

This is really an amazing technique. Thanks a lot. Might be useful in radial distribution functions to calculate potential energy of a radial field.

[@eduardodanielfarfanduran6346]

Alguien que pueda explicarme el porque de sinx=e^ix??...

[@artificialresearching4437]

Isn't it easier to get another expression of this integral using integration by parts and then deal with simple integrals of exp? But thanks for the video, I didn't even think of trick like that. Should learn more about that)

[@SKARTHIKSELVAN]

Thanks for your efforts in making these videos.

[@maxsch.6555]

Great video as always! :)

[@dgrandlapinblanc]

Excellent ! Thank you very much.

[@nchoosekmath]

1:58 "I am not gonna prove it" "I don't quite know the proof" lol

[@shapirogensichwa]

What are you expressing through this comment ?

[@nchoosekmath]

@@shapirogensichwa Nothing offensive actually. Just that when a professor says "I am not gonna prove it", I always think that it is tedious and irrelevant, so the audience should prove it themselves as an exercise. Turn out that it's hard and Dr Peyam did not know how to prove that.

[@Sitanshu_Chaudhary]

Why you make your video in the corner side of camera is someone is there to understand what you are doing however you are also fan of ramanujan

[@98danielray]

5:57 is that 2 references at the same time :v

[@shankarrahi2067]

Ramanujan sir ka unsolved problem ka solution mere pass hai and pi ka correct man bhi mere pass hai.

[@harikishan5690]

nicee🎉

[@motmot2694]

pi/2 became pi/4 magically 🤔?

[@motmot2694]

I think you meant pi/4. Great video nevertheless

[@98danielray]

wher

[@motmot2694]

Fractal 7:07

[@98danielray]

@@motmot2694 oh yes

[@nik_semperlotti1062]

I would say it's nice, but I need the demonstration to believe you.

[@akshatahuja2523]

Proud to be indian

[@jackrubin6303]

I don’t understand how in the step 2 part of the video for Ramanujan Integral you went from sinx to e^ix without considering the e^-ix component and 1/(2i) component. Sinx = 1/(2i)(e^(ix) - e^(-ix)). Also when I used WolframAlpha it gave a different answer to you. Respectfully Jack Rubin

[@shaochen5821]

This is a hard integral... Or is it?? *VSauce music plays*

[@erfanmohagheghian707]

We didn't need Ramanujan at all. J is simply Laplace transform of t^(-2/3) for s=1-i. As simple!

[@Sitanshu_Chaudhary]

Hee

[@Gamma_Digamma]

Wowie

[@Santoshkumar-qg4il]

Are भाई हिंदी मे बोलते तो अच्छा होता