Ramanujan's master theorem is insanely overpowered!!! example using the Fresnel integrals
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- Опубліковано 7 вер 2024
- The one theorem to rule them all.....
Ramanujan's theorem is one of the most powerful tools in integral calculus and today we're exploring a solution development for the fresnel integrals using this amazing result.
I think it would be better for the viewers who don't necessarily study pure maths's at uni to make an introductary video to some special functions and their properties and of course proof why it's working in the first place.
In addition to this a video covering the best pure maths books to study and/or course material recommendations would be nice, especially if one is wanting to look into higher math for an undergraduate’s
Ramanujan and Laplace are so intense they are the Double Rainbow.
Beautiful approach! Love Ramanujan. No one else like him
Ramanujan is unique, his mind so rich and beautiful 🥺, whenever I feel claustrophobic, He is a potent natural antidote.
bro just spawned in, dropped some absolutely crazy math shit, and dipped
@@orang1921pffft so true about that lmao 😂
One ring to rule them all one ring to find them one ring to bring them all and in darkness bind them
One Integral to solve them all 😎
Next video: drive the Master Theorem itself. This felt like a "shortcut" that required even more complicated assumptions than the method actually applied.
Thank you for this fruitful effort. The last step is the integration of cos(x^2) is with respect to x. So, dx in the LEFT hand side is missing. Thanks.
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I personally prefer the approach of complex analysis, you can take loop that depend on a variable R and the loop is basically a semi circle and just use the residue theorem. You'll just have to majorate 1 or 2 integral and it'll be done.
The melin transform and gamma function are both subjects of complex analysis rather than real analysis. So you can still count this video as one for complex analytic methods.
I like this one :)
And I am officially successful in summoning the ghost of Ramanujan
@@maths_505 lol 😂
My favourite way of solving these integrals is using contour integration.
The solution using contour integration is pretty cool and one of the more "unconventional" ways of designing contours. So yes I agree with you that the coolest way is contour integration (though for me the Leibniz rule ties it for top spot)
i wouldnt even think this integral converges
yeah I don't think it does
@@yuyan930t literally does. Otherwise he wouldn't have been able to evaluate it. Imagine saying the integral doesn't converge on a video literally solving the integral with valid steps. For a formal proof of convergence, you can just Google search.
What I find interesting is that the two integrals are equal. Equivalently, \int_{0}^{\infty} sin(x^2) \,dx = \int_{0}^{\infty} sin(x^2+\pi/2) \,dx, which by the way nicely corroborates the square root of pi/2. It's surprising when you look at the graphs.
I had the same thought when I first studied these integrals
@@maths_505 also if the limit is from -infty to +infty, you can do away with the 1/2 and the sqrt(pi/2) is then a straight consequence of the integrals being the same. The integrals being equal is a an equivalent statement to their value being sqrt(pi/2).
Where do I apply such masterpiece? What kind of problems? What kind of industries can be blessed with such knowledge?
Check out the wiki page of Fresnel integrals for that.
Amazing!
You can use the complex exponential form and solve the subsequent Gaussian integrals.
There are a couple of ways to solve those.
Feynman's trick
Laplace transform
This time I wanted to call the ghost of Ramanujan
After watching this solution you have more appreciation for Laplace
Laplace vs Ramanujan would be a nice integration war. I'll make a video on that soon.
@@maths_505 Feynman would dwarf either of them tbh
@@joshuahoover7700 the integration GOAT 🔥🔥🔥
Awesome ! I feel like a 5 year old discovering grown-ups toys !!
This helps alot! thank you sir!
I don’t remember the Mellon transform, it’s sounds like a calc 3 type of equation. I loved these long proofs in engineering school watching bleary eyed in class then banging out problems for the rest of the week.
Its Melin not melon 😂
@@maths_505it's Mellin, double "L"
The L was taken by the ones who doubted me yo!
(Yes this is a cover up attempt and thank you for pointing out the mistake)
convert to img and Re components, let u = sqrti(x) easy
Great video man
Thanks mate
When you are recording screen on galaxy tab, can we do youtube live at the same time?
how does this even make sense? aren't there infinitely many analytic functions that coincide at every integer (just think of adding sin(2piz) to w)? how is this well defined?
Sometimes, mathematics is about finding the best reason to bury your head in some sand and get some calculations done.
Hey may I ask what app you are using?
Samsung notes
@@maths_505 thank you
very cool
We have Fourier transform then Laplace transform and now even Mellin transform. Is there some other hidden transformation?
Z-transform comes to mind. They are all somewhat related to each other. I think the Z transform is used for sequences and, for engineers, for discrete signals.
@@Risu0chan How are they all related to each other?
by a change of variable and/or a transform of the function:
Laplace L{f} (is) = Fourier F{f} (s)
Mellin M{f} (s) = L{f(e^-x} (s) = F{f(e^-x)} (-is)
I'm not familiar with the Z transform, but the relationship is explained on Wikipedia.
In computer science there are more. They call them "correspondences" but don't quite formalize them into "transforms" the way mathematicians do (i think because they have so many languages/notations to choose from). One example is between Church's Lambda calculus and Turing's machines. You could also argue that each of these has a correspondence to complete Godel languages; all support derivations (programs) through conditional execution and arbitrary memory.
Curry-howard correspondence is an other big one. NOt an expert but suspect curry's work, in particular, has strong connections to Galois theory.
They have varying proximities to "more conventional" areas of mathematics.
I think the common underlying theme is that transforms/correspondences relate languages that can detect each other's errors. In that way they're quite like isomorphisms between "versions" of mathematics.
Nice!
can you do more videos on the melon transform?
Melin transform bro😂
If I find some interesting ones then yes sure I'd love to. You can find a table of melin transforms on the internet too
@@maths_505 ok perfect beautiful, gracias mija
.x^2=z, 2x dx=dz
Sir am ur fan and i need the inverse Laplace of 1 over [(square root of s)+1]
Plzzz
This isn’t google
@@johnbussio7742 you need to back up before you get smacked up
I think its t^(-1/2)e^(-x)/sqrt(pi)
@@maths_505 sir
I meant that 1 is not under the sqrt
that means 1/sqrt(s)+1
@@Mayk_thegoatfunny, but he has a point.
I think you made a mistake. When you replace the Us back to Xs, it should be x.e^(-ix^2). Or are you calling a completely different variable x?
Its just a rename. Not a substitution. The variable in case of a definite integral is just a dummy variable so you can rename it anytime. It doesn't change the functions or limits involved so it doesn't matter what you call it.
@@maths_505 I disagree... If it's just a rename as you claim, then this latter x is not the same as the first one, which you already declared equal to x^2. But if it is the latter x = u then it becomes confusing since you suddently have two different expressions for the I= where both include an x, but not the _same_ x. Conversely, if it is not really a rename then it is no longer the value of I being expressed. You do need to change the limits (both times you substitute), but incidentally x=0 u=0 and also x->inf => u=x^2 -> infinity. That might explain why one can "get away with" the "rename"...
@@benhetland576 That's what he said in the lecture. He mentioned that because the bounds remain the same, he can just rewrite a new function that is equivalent to the old function using different variables. That's like saying f(x) = x^2 and g(t) = t^2 (on same domain/range, etc). We can then say f(x) = g(x) = x^2 or f(t) = g(t) = t^2. There's no difference. When doing the integral, both integrals evaluate to the same outcome with the provided bounds, and thus we can do a change of dummy variables to get it in the standard terms of x again. Of course, @maths_505 is doing this in an 11 minute video instead of writing a whole LaTeX paper with all of the steps mapped out, he just glosses over the simple fact, and children in the comments section have to argue, even though they will learn how simple of a concept an exchange of dummy variables is in Abstract Math in First Year.
In his case, he is saying I(u=x) = I(x) = *integral with respect to x*
@@benhetland576 You need to understand the meaning of a mute or dummy variable. In this case u and x are dummy variable, and they can be renamed as you see fit.
are there any underpowered theorems?
Not on my channel😂
Rolle's theorem.
when the power series doesn't exist
But what if c-a-t really spelled, "dog"?
nice video u sound like andrew tate
After watching im starting to think that 1 + 1 doesnt equal 2
😂😂😂
@@maths_505I'm not getting it, can u explain please?
I clicked because of the clickbait title and video.
You kinda have to prove the theorem though
Bruh
That needs a video of it's own
But yeah that's a nice idea although I think Michael Penn has a proof video on this.
@@maths_505 it's just that there's a big difference between a video where I have to believe you and a video where I don't. My math journey is all from first principles, I have verified it all with my own eyes, yknow? I'll look for the Michael Penn video.
@@jkid1134that's great, but theres always too much to prove, since mathematical knowledge is so vast. That michael penn video is very nice tho.
@@lih3391maybe we just need shorter proofs. I think that when you approach a problem with tools it may accept but does not need, the solution is always harder.
I'm having a headache.....
Too much complexity for my brain....
Loool, i feel you.
it's a little more straight forward if you start at about 4 minutes. He explains the master theorem, which seems to define an other way to compute a mellin transform. Hopefully he does a derivation soon. Would be a lot easier for people who don't do the calculus every day to follow.
Ramajan said hhh sum of 1_1,,,,= 0.5 I think Ramajan stupid
and i think you are stupid
for an alternating series of 1 and 0 which is diverging due to the assumption of constant ( converging) series he does get the answer of 1/2 .
u can't simply say he is stupid.
by assuming the series of 1-1+1-1+1-1+...... converging
he got a value of 1/2
how in the world u can say it's stupid.
it's like a probability of any one answer that is either 1 or 0
Anyone suggest me how to remove this kind of worthless videos permanently...
😂 such worthless videos can never be permanently removed as long as there are worthless people who click on it, watch it and post worthless comments 🤣