Frullani's Integral Identity

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Today we be deriving Frullani's integral identity! :) We will need it for more analytic number theory identities and will use it for example, to derive a spicy integral representation for the natural logarithm, which is in close relation with the digamma function and harmonic numbers! :) Enjoy! =D
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[@NPCooking69]

> Grothensnecc is not real, it can't hurt u > 0:09 I hope you did enjoy this video, even doe I still feel a bit sick. Next video'll be better,, I swear! :)

[@matron9936]

You shouldn’t have had said snecc π times

[@twistedsector]

Papa, will you ever do something the Heegner numbers?

[@MarkusDarkess]

kzbin.info/www/bejne/kJ6uooKHj5p1o7c Can you help me rap my brian around orgin. 0,0. 4 3-D objects. Have an orgin. And 1 3-D Object. Doesn't. The 3-D object,I created it using pythagream thearum. But a 3D object with a 0,0 orgin cannot be created solely on pythagream thearum. And it is a rectangle on all six sides.

[@MarkusDarkess]

@Leonhard Euler I am using a touch screen and not a key board. And that is my access to the internet as a whole. 90% of the time. I apologize. And it is hard to make sure every word is always perfectly spelt. When going back and rereading. Because no one person knows how to spell all the words in their own language. Spelt and spelled are as correct as color and clour.

[@niccologeraci9175]

"I caught a terrible cold over the weekend". Papa has covid and still makes videos

[@PapaFlammy69]

Yeye, no way I would ever stop working!

[@sahilbaori9052]

Papa got covid? Seriously?

[@DendrocnideMoroides]

@@PapaFlammy69 you will continue working even after you die??

[@blackhole3407]

me: Ooh a new video! Casual intro happens, maths meme shows up me: yee its starting papa flammy: literraly breaks his voice chords

[@ricardoparada5375]

I’m liking these integrals with connections to analytical number theory

[@PapaFlammy69]

me too! :D Very satisfying identities if you ask me!

[@ricardoparada5375]

And very satisfying double integral too :D

[@orenfivel6247]

​ @Flammable Maths ,with loss of generality [wLOG] , i think also we need a,b>0 because if you replace boundaries" x=0 to ∞" by "x=ε to R", the f(0) part holds when ε→0, but f(∞) part does not hold because we may have f(tR) →f(-∞) when R→∞ if for example a

[@notabotta3901]

This first minute of this video was quite the wild ride

[@elib2670]

I just about stroked out when you came in godzilla mode

[@PapaFlammy69]

xD

[@natesavage3576]

Hmmmmmmm, let me integrate myself into this spicy comment section.

[@PapaFlammy69]

[@klofat]

thx, i spilled my drink

[@PapaFlammy69]

xD

[@caldera99]

What an opening sound! Incredible!

[@IlTrojo]

Nice tricks, but you can also: 1) consider a as fixed value and an equality in b; 2) verify that the equality holds when b=a; 3) take a Feynman derivative w.r.t. b on the left and a regular one on the right and see that the two are indeed equal; 4) QED.

[@kazimendez461]

Love this channel💫

[@MrRyanroberson1]

this is such an interesting idea, i can't wait to see what you apply it to!

[@PapaFlammy69]

:)

[@parameshwarhazra2725]

Sir I just got really amazed when I saw the notification. Didn't think of anything and click on the video in hurry. With love from India

[@PapaFlammy69]

:)

[@b1zzler]

0:42 - the new official sound of mathematics 😌

[@PapaFlammy69]

:D

[@hernanfelipegonzalezaguirr7802]

I do not want to be the virgin Chad, but we must assume f is continuous to put the limit inside the argument, so the formula must have the limit out the f. That's all :) Great content papa, as usual !

[@let1742]

I may not understand all of this, it's incredibly pleasant to watch, thanks for your work !

[@s1ng23m4n]

Not so long ago I learned this method. A half of all my problems can be solved by this method or similar one) Nice.

[@HAL-oj4jb]

Holy shit the grothensnecc thing actually worked

[@jony7779]

Love this Brah!!

[@gamingstars8956]

I love integrals🤯🤯

[@PapaFlammy69]

me too :3

[@neilgerace355]

0:25 Nah, the worst is "if you trust the Peano axioms" :)

[@PapaFlammy69]

But... but Neil ;_______________________________________;

[@patricius6378]

HYPED for epic monster slaying although I don't know calculus

[@tomasblovsky5871]

I know this beautiful formula from a book called Exercises and problems in mathematical analysis by B. P. Demidovic. Do you know this book? You can search just Demidovic, its a red book. Theres a joke when people say that if you solve all the problems in it, then you are better than your math teacher :D Anyway great video as always

[@PapaFlammy69]

nah, didn't know about it! =)

[@tomasblovsky5871]

@@PapaFlammy69 Oh, then Im glad Ive told you. :) Here, in Czech republic, it is a very recommended book for exercises haha. The book can be downloaded of course :D

[@brendanlee1707]

Respect papa

[@zanerobison2956]

I love your channel LOLOLOL

[@johannesbartenschlager3426]

You need a little more care at the integration by substitution at 4:03. At the left hand side integral you need a factor x for the boundary.

[@johannesbartenschlager3426]

For the justification of the interchange of integration, since the limit for x to infinity must exist and in order to have a derivative, f needs to be bounded, so just add a constant C large enough such that f+C>0, then use Tonelli's Theorem.

[@frozenmoon998]

Axiomatic approach to snecc theory: *Denote 57 as prime, bow down to his majesty the snecc, subscribe to Papa Flammy now or you are going to become Presh Toadwalker*

[@mohammadburhanmanzoor3806]

@4:09 the limits of eta are ax to bx, obviously. But well engineer mode so coolio papa! ahhhhhhhh!

[@NPCooking69]

You are right! Nice catch, I honestly did not notice! :D

[@mohammadburhanmanzoor3806]

Flammable Maths Two 🙃

[@BardaKWolfgangTheDrug]

So basically this video is 0:50 - Papi showing his muscles 3:00 - Papi showing his artistic skills 6:16 - Papi showing his Kittie Catties Afterwards - Papi showing some ez maths. I feel confused when the content is easy enough to understand it besides being focused on his muscles instead of the integrahl 🙈

[@PapaFlammy69]

:'D

[@kiiometric]

I thought about e^-x even before you said it, what is this power I'm feeling...? PD: great video Papa :D

[@Galileosays]

Nice. Thanks to Frulliani, who derived it at age of 26. He died at age of 39 in 1834 in Florence.

[@yassinezaoui4555]

Do know guys any app in the play store that allows me to write math equation,... and then turn it into a picture? Same question for the pc, is there a software or sth ? Btw, nice vid man ^^

[@sunandobanerjee7665]

actually integration of g(tx) dt from a to b will be 1/x times integration of g(u) du while u from ax to bx and not a to b. But in end if you substitute u as tx, then the original lower and upper bound values can be restored, I was confused when you didn't change the upper and lower bound a and b to ax and bx during the observation.

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I think your muscles ( biceps ) are also better day by day. Keep it up . Looking nice😃😃

[@PapaFlammy69]

haha xD

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Any chance you would make a video with your shirt off?

[@integralboi2900]

By 2021 Papa will have made a Snecc religion.

[@PapaFlammy69]

:'D

[@jkid1134]

9:52 look at this cursive ]

[@nathaniels8734]

Does this fly for complex a & b?

[@gregoriousmaths266]

Lmao grothensnecc 😆😆😆😆😆😆😆😆😆😆😆😆😆😆🤨😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆😆

[@sansamman4619]

are we just going to ignore how muscular papa is?

[@PapaFlammy69]

yeye my dear San :p

[@PapaFlammy69]

what a great word play xD

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HELLO SIR I AM YOUR HUGE FAN LOVE FROM INDIA❤️❤️

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[@pmcate2]

How do we know that f(0) and f(inf) exist? Or would we just conclude that the integral diverges if these don't exist?

[@angelmendez-rivera351]

If the integrand satisfies Tonelli's theorem and Fubini's theorem but f(0) and f(♾) somehow still do not exist, then yes, we would say that the integral does not exist (diverges).

[@calcul8er205]

pmac there are some integrals of this form where f(∞) diverges yet the integral converges. f(x)=sinx and f(x)=cosx for example oscillate forever but the integral converges. This can be shown by introducing a normalising factor of e^(-sx) then letting s approach 0 or through other means

[@calcul8er205]

This result can also be used for f(x)=sinx/x but obviously you need to define f(0) and f(∞) using limits

[@duncanw9901]

Nightcore vocals be like: 0:42

[@torment808]

maths is like “simple proof” *proceeds to pull out polygamma function* *proves pi to be irrational*

[@zarifahmed8549]

Where do you study man?

[@sahilbaori9052]

Why is 57 the best prime tho?

[@alexismiller2349]

2:35 shouldn't the lim be outside the f(), not inside?

[@PapaFlammy69]

If f is continuous, then it wouldn't matter^^

[@alexismiller2349]

@@PapaFlammy69... mmh yes I would agree with you if x was converging to anything else than infinity, but like this it looks weird to me, idk if it's just a convention because otherwise f(lim x) = f(infinity) doesn't seem right

[@angelmendez-rivera351]

alexis miller No, it still is correct when x -> ♾. It does not matter what x is approaching: a function and a limit are interchangeable if and only if the function is continuous. Actually, this is the definition of continuity.

[@tetraedri_1834]

@@angelmendez-rivera351 Only if the limit we are taking is inside the domain of the function. In this case, I guess f would be defined on reals, and infinity is not a real number, so the limit f(x->infty) makes no sense. An easier example: consider the identity function id:(0,1)->(0,1) of the open interval from 0 to 1. Then id is clearly continuous and lim(x->1)id(x)=1, but 1 is neither in the domain nor codomain of id, so id(1) is not defined.

[@angelmendez-rivera351]

Tetraedri_ *I guess f would be defined on reals, and infinity is not a real number, so the limit f(x -> ♾) makes no sense.* It makes perfect sense. Infinity is not a number, and so the definition of evaluating f at infinity is evaluating the limit of f as x approaches infinity. This is true regardless of whether f is eventually continuous or not: if f is not eventually continuous, then the limit does not exist to begin with. *An easier example: consider the identity function id : (0, 1) -> (0, 1). Then id is clearly continuous and lim id(x) (x -> 1) = 1, but 1 is neither in the domain nor codomain of id, so id(1).* Not only is your argument incredibly sloppy, but also fallacious even in the would-be case of it not being sloppy. 1. id is only continuous in the interval (0, 1), because by definition, the domain is (0, 1). It is *not* continuous at x = 1 like you claim. Also, lim id(x) (x -> 1) does not exist, because lim id(x) (x -> 1+) does not exist. It would be more accurate to state that lim id(x) (x -> 1-) = 1, but this is far from sufficient for id(x) to be considered left-continuous, let alone simply continuous. 2. Even if you had not been sloppy with your claims, you still wasted your time with your argument, because your argument is a false equivalence. You are trying to present it as some example, but the reality is that this not an example of what we are discussing, and so the comparison is invalid. That is because 1 is a number, and ♾ is not. It makes perfect sense to talk about lim f(x) (x -> 1) and f(1) as being separate quantities, because 1 is a number, and you can evaluate functions at a number. ♾ is not a number, and so the notion of evaluating a function at ♾ makes no sense. However, it does makes sense to talk about limits with ♾, so colloquially, we take "evaluating f at ♾" to be a notational shortcut for "taking the corresponding limit." There is no distinction because by definition, there cannot be any. f(♾), lim f(x) (x -> ♾) and f(lim x (x -> ♾)) all mean the same thing by definition because it is impossible for them to mean different things.

[@blackhole3407]

What am i doing here, im a 14 year old pianist...

[@Observer_detector]

what a weird result! it is so weird XD

[@hoodedR]

Snek Snek Snek Snek Snek Snek Snek Snek Snek It didn't work 😢

[@PapaFlammy69]

where ur mirror doe? Damn, feels good to finally find a bit of time to heart and reply again! Thank you for your membership support Ranjan! I really appreciate it!

[@hoodedR]

@@PapaFlammy69

[@PapaFlammy69]

[@Happy_Abe]

Why does f(0) need to equal 1?

[@integralboi2900]

It’s easier to have a function where at x=0 equals 1, and approaches 0 as x approaches infinity since we know basic functions like e^-x would satisfy that.

[@PapaFlammy69]

ye

[@Happy_Abe]

@@integralboi2900 but can we in theory construct a different one?

[@integralboi2900]

Avraham Radin of course!

[@Happy_Abe]

@@integralboi2900 ty

[@xbowschool]

Does anyone else realise 57 is not a prime

[@tatjanagobold2810]

What if I say Flammy into a mirror tree times? :o

[@twistedsector]

Nice, just say snecc pi + e + sqrt(g - 1) times

[@dimdiemer3078]

nobody going to comment on whatever emotion 0:43 was?

[@divyanshichauhan4408]

I am the first viewer I have done it🎉🎉 Edit: I am always excited to become first viewer of a video😎😎

[@PapaFlammy69]

ez

[@MathIguess]

I'm gonna send you an email after posting this comment :p

[@adityaruplaha]

Holy shit I got a seizure when you did that 0:43

[@beforemidnight9938]

that intro shreek was lowkey kinda hot ngl

[@JoeFru]

My last name is Frullaney (formerly Frullani) 🤔

[@neilgerace355]

7:11 Just Fubini the shit

[@thephysicistcuber175]

7:10 Fubini requires absolute integrability.

[@davidepierrat9072]

The Physicist Cuber u r everywhere

[@thephysicistcuber175]

@@davidepierrat9072 Exactly. "Everywhere" is my 76th name.

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@@davidepierrat9072 also nice, you watch papa flammy.

[@davidepierrat9072]

The Physicist Cuber ofc

[@Ryan-gq2ji]

@The Physicist Cuber what do you mean by absolute integrability if you don't mind me asking?

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Nice arms dad

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Happy Cirno Day!

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Zfthank you

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Snec snec snec

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WOW

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69th

[@PapaFlammy69]

nice.

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20π-e/sqrt(g)th

[@devanshsharma8638]

First

[@EssentialsOfMath]

Sorry bro ur two years late

[@michaelg5932]

First

[@valovanonym]

Papa Flamy was here before you.

[@michaelg5932]

@@valovanonym true, he always comes first😂

[@valovanonym]

@@michaelg5932 ofc 😅