γ - A BRILLIANT Putnam Integral Journey - Analytic Number Theory at its finest

[@davidalexander4505]

Pure math student here. We're allowed to do this by what's known as the dominated convergence theorem. This theorem is equivalent to saying that you're allowed to bring a limit from the inside to the outside the intrgral provided that the integrand converges uniformly to some continuous function and the outside limit exists. In our case, the convergence is uniform as the integrand viewed as a function of n is a continuous function on a compact interval [epsilon,n] for all epsilon > 0. The integral is known to exist and so, limiting epsilon to zero gives the correct result in this case. Also, letting n be the upper bound is also totally fine if you know that the limits exist independently, which in this case, they do. I've skipped a few details but the central idea is the dominated convergence theorem, double limits existing implies you can calulate them using convenient 1 dimensional paths, and integrals existing implies that integration over a limiting boundary will yield the same result.

[@davidalexander4505]

@@PapaFlammy69 that's all good, I love your videos man!

[@akeelhowell8944]

David Alexander was basically going to say this, but decided to see if anyone else did first

[@alicewyan]

Gosh, thank you, the upper bound substitution was puzzling me!

[@kummer45]

These are the details that needs to be said for those who wants more rigor on the derivation. His approach is acceptable giving an heuristic introduction. However a written comment on certain items presented on the video may furnish a solid base on these matters. Excellent observation Alexander.

[@AndrewDotsonvideos]

This.Is.So.Oily leave an Euler if you macoroni’d😫😫🤷🏿‍♀️

[@austinmitchell2652]

e

[@alse72]

Mas"k"eroni please, not ma"sh"eroni. Also the limit notation as "سا" should be taught in schools

[@neilgerace355]

What does the Arabic you have written really mean, please?

[@alse72]

Sorry for spoiling it but it's just the letter s followed by an a, so it's just "sa" which doesn't really mean anything as far as I know

[@rot6015]

mans fun at parties

[@alse72]

@@rot6015 you can't even imagine ;)

[@susiehue9465]

I disagree. Arabic letters and numbers should not be taught in school. We should teach our children good o'l Christian numbers

[@MrLikon7]

wikipedia: The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational.

[@zoltankurti]

That's what you get if you come up with weired numbers!

[@createyourownfuture5410]

It has to be irrational... By intuition If you define numbers by weird functions must likely it is irrational.

[@DendrocnideMoroides]

@@createyourownfuture5410 it is almost certainly irrational but there is no proof of it

[@sudhanshumishra6482]

Really love watching you solve integrals.

[@PapaFlammy69]

:))

[@rezomegrelidze7748]

Didn't know Marco Reus was a mathematician.

[@Mindcrackings]

Can't unsee lmao

[@hal6yon]

So that's what he was doing while bayern ripped his team a new one

[@peppybocan]

Oily-Macaroni!

[@neilgerace355]

But ketchup and mayo? Who does that??

[@peppybocan]

@@neilgerace355 you do that for burgers. You mix up mayo with a little bit of ketchup (and even mustard, if needed!)

[@neilgerace355]

@@peppybocan but macaroni! As an Italian, I'm pretending to be offended :)

[@neilgerace355]

Oily macaroni is what you get if you use the wrong cheese

[@peppybocan]

@@neilgerace355 yes. Very fat cheese.

[@Gustavo_0107]

Make a video solving that monster from april fools day!!!!🌋🗺🧭⛰⛰🌍🦓🐈🐎

[@okoyoso]

Apparently, the notation for lim in arabic is نهــــــــــــا so that's not far off lol

[@Ricocossa1]

Yay! The Oily Macaroni constant! By the way, Mascheroni is pronounced "Maskeroni" (Italian 'ch' is 'k')

[@tszhanglau5747]

Oiler-macaroni with ketchup sounds like a good topic for a cooking video...and the manipulation of the bounds is awesome.

[@neilgerace355]

14:26 White line fever, as we say in Australian football

[@theflaggeddragon9472]

Regarding analytic number theory, would you do a video on the derivation of the functional equation for the Riemann zeta function? I really don't think it's more difficult than some of the crazy stuff you've done, and it's crazy useful in big boi maths

[@mihaipuiu6231]

I watch you for a few years, because not only that you are a very good mathematician,...but b/c you are a good speaker, with writing very clear, and nice 'n neat. Also, you are very FUNNY, and this is always a plus. Congratulations!

[@PapaFlammy69]

[@TimesOfSilence]

I was really verwirrt at the beginning, because there were no Klammern between the Integralzeichen and the dx. So I thought, that would end in really crazy mathematics, as the Integralzeichen only belongs to the first part of the sum, and the dx only to the second o.O

[@hal6yon]

He is speaking the language of gods

[@Ukrainewinnerrrr]

Lmfao der hat Recht !

[@jkid1134]

Name a more iconic duo than papa and gamma, I'll wait

[@Reliquancy]

It seems easier to just expand integral from 0 to 1 of 1/(1-x) as integral from 0 to 1 of sum from 0 to n of x^k. then swap the sum and integral so you have sum from 0 to n of integral from 0 to 1 of x^k. the integral becomes x^(k+1)/(k+1) by power rule evaluated at x=1 minus evaluated at x=0 gives 1/(k+1) which you sum from k= 0 to n.

[@looiepoohie7363]

Papa flammy! You requested my response to "Why does Automatic Differentiation using Dual Numbers 𝑑=𝑎+𝜖𝑏,𝜖^2=0 work so absolutely amazing?" on Quora. 1.) I didn't know you had a Quora! 2.) I have no fucking idea how to answer that. Glad to see you on other intellectual platfroms. :)

[@Riiisuu]

This better come up on my first year uni maths exam or I am screwed

[@jorgemedrano9132]

"We can do this if this converges, and it does" proof by intuition

[@angelogandolfo4174]

14:30 - “no, it’s because of the chalk dust, I’m terribly sorry....” yeah, yeah, sure it is...... we ALL believe you, honestly...... 😂🤣😂😤🤧

[@neilgerace355]

7:30 But from left to right, just to be fun 8:07 When in doubt, just Fubini the shit

[@emilyscloset2648]

Now I'm down for some oilly pasta. But in all serious that was fun to watch. Props

[@mrandersonpw53]

About the step in 9:30, (1-t/n)^n / t. can't tell why you can do that. The function is not L1, but (1-t/n)^n / t seems to increase for t>1 fixed. Must be by Fatou's Lemma plus the vertical truncation for N=max{t,n},0=< t=

[@Uni-Coder]

Will any mathematician blogger ever make a video about probability theory? Then, mathematical statistics, and then, data science and machine learning, and then, artificial intelligence! Very modern topic

[@erikdurfey5576]

That in-te-ge-ral sign at 4:56 tho

[@aboutmath2995]

great video man!

[@martinepstein9826]

Awesome integral but I'm suspicious of the method. Your first step using linearity is illegal since both of those integrals diverge. To split it up you'd first need to take the integral from epsilon to 1 - epsilon and then take the limit as epsilon goes to 0. Then by making different substitutions in the first and second integrals the bounds become different functions of epsilon and you can't use linearity to recombine the integrands.

[@cedricp.4941]

Integral from 0 to 1 of 1/t dt = pi.

[@martinepstein9826]

@@cedricp.4941 Good to know.

[@twistedsector]

He's a physicist what do you expect

[@alicewyan]

physicists are allowed to split divergent integrals as long as the divergent parts cancel out in the end ;)

[@tshepisomhlanga5389]

This is one interesting intehgeral

[@sansamman4619]

I'm still watching all your videos but i no longer have the extra time to comment love you papa

[@wolframhuttermann7519]

Man kann sich das leben komplizierter machen, als es ist. in 16:32 kann ich auch für x = 1-(1-x) im Nenner schreiben und dann vermeidet man eine weitere Substitution.

[@snfn7847]

Gucci video papa

[@Павел-и5б3ц]

Классное объяснение. Спасибо.

[@gesucristo0]

how do you show the limit doesn't diverge?

[@kummer45]

I'm not going to be very constructively critical about these series of videos. You are very responsible saying the necessary theorems involved and showing the needed steps to arrive at the solution. I've noticed that you are generous providing the technicalities too. You answered your call and have all the talent needed to become a good teacher and instructor. Math always happens as a problem solving endeavor. My critique or observation in this case is more of a recommendation. Some sources should be quoted so those interested on other details may access the information. For example saying that such topics belongs to "SPECIAL FUNCTIONS" and "Fractional Calculus" would help people where to dive their curiosity. Your approach is interdisciplinary to the point that requires knowledge on few subjects.. When I make proofs I quote the source or the author that I read beforehand so people follow or read more the sources on the subject. Mathematics IS hard but it doesn't mean that mathematics stops being beautiful. People MUST realize that math requires ALWAYS paper, eraser, pencil in hands ALL THE TIME. When I construct the derivations there are few steps that asks for the gymnastics of algebra and the theorems. I know that you intend to do it user friendly under a span of time but when people dives their nose into the details, all of them realize that it's full hard work and LONG hours understanding what is GOING ON when these derivations happens. In general I don't see any other issue with your presentation. They are very good and on point. For further details people should fill in some details if they want more insight of what is going on. :P tyvm for this content.

[@pythagorasaurusrex9853]

Ahhhh! Ich habe es eigenständig bis 6:30 geschafft, habe aber dann versucht mit der Taylerreihe von exp(t)/t weiterzumachen. Bin danach nicht mehr wirklich weiter gekommen. Mir kam zwar dann die entstehende Reihe "irgendwie" bekannt vor und der Taschenrechner hat dann nach ein paar Iterationen 0,577.... ausgespuckt, ich habe die "Macceroni Alio-Olio"-Konstante nicht gesehen. Daher.. Daumen hoch für (1+t/n)^n als Alternative zur Taylerreihe. EDIT: Noch was. Am Anfang des Videos sagst du, dass du nicht weißt, ob Gamma irrational ist oder nicht... Haha... Das weiß zur Zeit niemand!

[@user_2793]

I literally woke up in the middle of the night thinking about this : What is -1 raised to an irrational power?

[@angelmendez-rivera351]

-1 = e^πi, so (-1)^x = (e^πi)^x = e^(πxi) = cos(πx) + isin(πx). For example, (-1)^π = cos(π^2) + isin(π^2)

[@Toxic__rl]

love your vids, btw, is it okay not to put (...) before d(something)?

[@FT029]

13:43 I'm not entirely convinced you're allowed to just add infinity (since the integral is ln(n)) and subtract it. Or perhaps I'm missing something simple.

[@oni8337]

he said it does exist

[@matheus_rml]

This constant is almost sqrt(3)/3

[@letolion]

8:48_These Sango_Kou Kameh -a_Meha moves lol

[@Gustavo_0107]

Yo is that true? I heard some people saying that germany will change its official language to english

[@abdullasulfikkar5282]

Mascheroni or macaroni????😂😂😂😂

[@danibarack552]

how can you make k and n look so similar?

[@EnginAtik]

Integral variable ‘t’ and limit variable ‘n’ are independent of each other. Did we assume a dependence between ‘t’ and ‘n’ while replacing the upper bound of integral over ‘t’ by ‘n’?

[@theoleblanc9761]

Rigor is dead. You can't split a convergent intregral into 2 divergent integrals

[@angelmendez-rivera351]

You can, because of Cauchy principal values.

[@zapuer1708]

Euler macaron

[@sahilbaori9052]

How can I share problems with you so that it may potentially become a topic of your videos?

[@ArthurvanH0udt]

AT 1m57 you say: "they both diverge so slowly that we get a finite difference"! Isn't that missing a statement like that they also both diverge along a common divergence line (sorry i'm not a pro mathematician but this is how I say it). I mean both could diverge very very very slowly but still both be/live in some different "locality" of infinity en thus the difference would/could also be infitie?

[@sanjusanju3454]

I like ur vedios make more about new topics in mathematics

[@goatmatata2798]

Nice watch =)

[@anthonygreven2811]

at 11:43 he wrote that "(dt-[1-t/n]^n)/t" is equal to "(1-[1-t/n]^n)/t"... how can we demonstrate this?

[@louisvan4395]

brilliant video! I have one slight question though. don't we need to make sure the function 'tau to the k' is uniform convergent before interchanging the integral and summation signs?

[@louisvan4395]

Also I love that you have videos both in English and German. I once visited Germany, and it's hands down one of the best languages I've had the chance to learn. I'm a proud polyglot myself, and I like what you've done with the channel

[@scottwhitman9868]

Its a finite sum

[@sounapet795]

i have one problem about intergral. intergral of x^8/cuberoot(1+x^4 ) please help me this one

[@desertrainfrog1691]

No elementary antiderivative.

[@ketermeissner9898]

'Maskeroni', not 'Masheroni'

[@verainsardana]

At 10:19 how can you say that limit infinity of integral is equal to that of definition of e, either infinity can be greater.

[@jameswilson8270]

Love your channel. But the very first step produces the sum of two divergent integrals. The Clay Mathematics Institute would have thrown you off the stage. ;)

[@jameswilson8270]

@@PapaFlammy69 It's all good, my man. Divergent boys are a pain in the neck.

[@zoltankurti]

How do you do 2:25 rigorously? In the video you got 2 diverging integrals and manipulated them separatly.

[@desertrainfrog1691]

Assume it works and then either be sad or happy depending on what happens.

[@zoltankurti]

@@desertrainfrog1691 Are you a physicist?

[@desertrainfrog1691]

@@zoltankurti I'm just a guy who's okay with things working out, whether they're justified or not ;D

[@zoltankurti]

@@desertrainfrog1691 hmm. Now that I think about it you could be an engineer too. Hope that's not insulting to you, I don't mean to.

[@sayantanmazumdar9371]

same question

[@non-inertialobserver946]

Integaral

[@pratik_shrestha]

Cicada?

[@ice-bug465]

What is the music that you use in the intro?

[@MuitaMerdaAoVivo]

i love how he says "we can just fubini this shit, probably"

[@Nick-hc2cs]

17:11 lel

[@general9064]

t = 1-x and t = ln(x) then you add them the integrals later. I don't quite get it

[@angelmendez-rivera351]

jeremy brett The dummy variable is irrelevant

[@hafarov8205]

Papa, I don't understand. First, you broke up the integral, then, you substitute -ln(x) with t, moment later, you substitute (1-x) with t. And, in my humble opinion, these two t are different. And at 11:35 you bring two integrals with two different t together. Isn't it a crime against Math or have I missed out something?

[@martinepstein9826]

The variable name in a definite integral doesn't matter. The definite integral is a property of the function being integrated, not the variable we notate it with. If two definite integrals have the same bounds then we can add them by integrating the sum of the functions in the integrands since the integral is a linear operator. So really it didn't matter that both integrals had the variable "t". That makes the proof easier to follow but it's no more valid than saying int[a,b] f(x) dx + int[a,b] g(y) dy = int[a,b] (f+g)(z) dz

[@hafarov8205]

@@martinepstein9826 oh thank you

[@tianyi1240]

Why didn’t you just write the first integral directly as \int_0^1dx/x? You would have saved one useless change of variables,

[@mohkerhakon5941]

int from 0 to 1 of int from 0 to 1 of (1-x^y)/1-x dydx=0.577215664...

[@ErhardNeher]

Limiteroid

[@shanmugasundaram9688]

You substitute t=- ln(x) for the first integral and t=1-x for the second integral.The t's in the first and second integral are different.

[@DragonKidPlaysMC]

Was thinking the same, but why?

[@dagkouta986]

7:02 did he.. did he just say anal 1 notes instead of analysis 1 notes? wow just realised how dirty math sounds

[@Ukrainewinnerrrr]

I do this calculus in 3 minutes ! Concerning the constant please say MasKerrrronnniiii per favore ! Merci !

[@bernardoxbm]

I would like to watch this video in German so I can practice the language.

[@bernardoxbm]

@@PapaFlammy69 ja danke

[@midas-holysmoke7642]

This video is nothing but...

[@muhammedalshaer3333]

These apples explanations don't make a shit of sense 🤣

[@muhammedalshaer3333]

Oh shit one of my math idols replied to me

[@theunfightable4513]

Viel zu umständlich geht leichter wenn du das e Ding direkt als Summe schreibst.

[@BhanuNarra1]

1000th view lol

[@BhanuNarra1]

Papa, can you do more Quantum Field theory

[@nickgibson3451]

When you had that small schpeil about getting to e^t = 1/x literally so many things made more sense, i dont know it may be bc I am a science undergrad starting an engineering phd, but I never had it explained and jesus christ it was so simple yo boy didnt even write it down🥲