The most beautiful result in calculus

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Maths 505

Күн бұрын

Пікірлер: 52

@m.m4217 10 ай бұрын

You could also take 1/2 inside the logarithm to obtain a square root and a more compact result

@lukeskywalker2255 9 ай бұрын

It wouldn't be as nice though

@jaafars.mahdawi6911 10 ай бұрын

I'm not as much of an integral guy as you are, but such elegant, meticulously crafted results do integrate the subject into a source of joy for me! Good job!

@maths_505 9 ай бұрын

Thanks

@mcalkis5771 10 ай бұрын

Hey man, I don't know your name, but I really appreciate all the work and care you put into your videos, they are always a nice break from my heavier university stuff. I hope you will continue your tensor series (possibly even your analytical mechanics one if that was ever going to happen). Also, could you do a video explaining when and why we can change the order of summation and integration in infinite series?

@orionspur 10 ай бұрын

Lordy. Amazing!

@sairux7960 10 ай бұрын

amazing.

@manstuckinabox3679 10 ай бұрын

Ah, it's always nice when the problem is actually a variation of some weird problem you solved a long time ago... The video should be called the equivalence class of the most beautiful result in calculus. Awesome work as always my friend.

@Calcprof 9 ай бұрын

Mellin transform was always the most fun part to teach in a graduate course on Integral Transforms. You could also get the series for 1/(1 + e^-x)^2 by the binomial theorem, and then work out what the binomial coefficient (-2, k) is.

@gutentag1752 10 ай бұрын

I think the Integral is known as one of Malmsten's integrals. But I am not sure if it has a special name like Vardi's integral which is also one of Malmsten's. They all have really fascinating results. It's really nice that you made a video about it.

@MrWael1970 10 ай бұрын

Thank you for this video. To be a complete documentation for this innovative integral, Please give the links for the proof of eta, first derivative of eta and gamma, for the non specialists and under graduate students. So, you are talented in solving such type of integrals.

@ruffifuffler8711 3 ай бұрын

Likely a seminary trick; 2 piggybacks go through the pi hole, probably get en-natured, then are not split, but partitioned into conjugates, which produces life.

@The_Shrike 10 ай бұрын

Drop merch and I’ll be the first to get some

@barryzeeberg3672 10 ай бұрын

Since e appears within ln, it seems that this expression could be re-written without the e . So it may be artificial to claim that e is wrapped up in the solution. i.e., take the expression "ln(e)". this just equals 1, so it does not really include e .

@Noam_.Menashe 10 ай бұрын

Well you could say a natural log is also jsut as special as e I guess.

@barryzeeberg3672 10 ай бұрын

well ln() is a function, not a special constant, and I think the author's idea is to get a conglomeration of several special constants.@@Noam_.Menashe

@joniiithan 10 ай бұрын

Legen -wait for it- dary!

@shivanshnigam4015 10 ай бұрын

Barn door, Stinson natti, bro hio

@joniiithan 10 ай бұрын

@@shivanshnigam4015 nice one

@joniiithan 9 ай бұрын

@@shivanshnigam4015 Bro I just watched some episodes and found another nice quote even regarding math: Barney: If we analyze the seemingly random patterns of the train, taking into account standard deviation, and assuming that epsilon approaches zero as angle delta approaches pi, we can conclude... Ted: [snores] Barney: Damn it, Ted! I was about to drop some sweet word play about logarithms and getting into a rhythm with my log. I'll remember it.

@shivanshnigam4015 9 ай бұрын

@@joniiithan yeh I've seen that too, they were in the suburbs at Lily's home 😆

@renerpho 5 ай бұрын

That's a great result. It would be even more awesome if you didn't have to invoke "log of e^something". Do you know of an integral (or a sum) that has e, pi and gamma, but with no logarithms in sight?

@aomaik7639 10 ай бұрын

Good work , can you solve this integral 1/2s integral 0 to infinite of x^s/cosh x -1 to get zeta and gamma functions 😢

@mikelevels1 10 ай бұрын

Bad ass!

@maths_505 10 ай бұрын

Hell yeah!🔥

@aidenmcdonald5605 10 ай бұрын

i wonder if you could alter it slightly somehow to involve the golden ratio too

@maths_505 10 ай бұрын

Wow now that sounds ambitious

@Noam_.Menashe 10 ай бұрын

@@maths_505 I think you can, but it will probably make it so that there isn't an Euler Mascheroni.

@illumexhisoka6181 9 ай бұрын

But the eta sum in this form doesn't convergence for s=1

@shivanshnigam4015 9 ай бұрын

Hey please try this one also Integral from 0 to 1 of (x {1/x}[1/x]) Where [•] denotes floor function and {x}=x-[x]

@shivanshnigam4015 9 ай бұрын

Bro can you make a video on this one

@carlobenedetti2407 10 ай бұрын

Did you find the series and integrals you show in your channel by yourself? Love your channel 👏🙌

@maths_505 10 ай бұрын

I got the series by playing around with the weirstrass definition of the gamma function. I also came up with the integral (the one after my substitution) while playing around with the melin transform I derived in another video. The integral can be found in tables like the one of Prudnikov but I haven't seen the series anywhere else yet.

@maths_505 10 ай бұрын

I've come up with quite a few of the integrals I've solved on the channel so far. Mostly by accident 😂 by messing up the solution of some other integral or experimenting with functions and limits.

@sharpnova2 10 ай бұрын

@@maths_505nice, weirstrass comes up a lot between me and my students. very useful everywhere nondifferentiable function for real analysis proofs

@lmaorofl3229 10 ай бұрын

Hi, I want to ask about the geometric series cant I just square the whole summation? I know its going to be impracrical but I am asking if its fine to do the squaring in normal cases. Sorry if I am fully wrong Im still learning I have to wait till high school

@maths_505 10 ай бұрын

Not exactly a good approach given the RHS is an infinite series.

@comdo777 9 ай бұрын

asnwer=1>1/2 os isit

@Noam_.Menashe 10 ай бұрын

This is a case of Malmsten's integrals no?

@maths_505 10 ай бұрын

The structure does agree with that

@insouciantFox 10 ай бұрын

The integral 0 to 1 of ln²(1+x)/x without Feynman integration has defeated me.

@Noam_.Menashe 10 ай бұрын

+I think the easiest way to solve it would be series expansion + cauchy's product.

@miguelcerna7406 10 ай бұрын

@alielhajj7769 10 ай бұрын

It’s a fake e, cause it can be eliminated by the logarithm

@maths_505 10 ай бұрын

Ridiculous! e is a member of the set of real numbers! How can you get more real than that!

@renerpho 9 ай бұрын

@@maths_505 What they mean is, you can always introduce an e into your formula, using the logarithm. The goal should therefore be to find an integral that evaluates to some combination of e, pi and gamma, but without logarithms.