Proof of Euler's Integral Formula (Gamma Function)

[@akashagarwal5215]

One of THE most beautiful proofs for the gamma function...just loved it...I usually don't subscribe to a channel after watching just one video but you deserved it

[@gustavosouza5600]

wow, i was looking for a proof for this integral for a long time and you gave me a neat one. thank you

[@henrykwieniawski7233]

Finally! I’ve been searching all over for a proof. Very well done!

[@gian2kk]

Been looking for this for years! Thank you very much!

[@chromiyum6849]

ABSOLUTELY mind blowing! Thanks for this

[@oakleymc9300]

Mistake in the opening screen; e^{-nx} should read e^{-x}.

[@michaelliu6323]

same finding for me

[@spdas5942]

Sir, you boosted me up by proving the gamma function and n! in this unique and understandable way. Hat off you ! ❤ from India. Remain healthy and vital.

[@hardcorelevelingwarrior6324]

Excellent proof, well explained!

[@kevinsommerfield6341]

This was a good proof, but there is an error in the first slide. You have written e^(-nx) where you wanted e^(-x). I wouldn't nitpick, but I just don't want anyone to be confused. Again, nice proof!

[@34sonamrani45]

kzbin.info/www/bejne/oXLLaGyoltOFsJY

[@nathandyson3462]

Thank you for this! I was was wondering about this for too long!

[@Huehhee7777]

Great explanation! I really needed this

[@monochr0m]

Great video! :) it would be nice to see just what happens when n is actually from R or even C instead of a natural number

[@LearnMathsFree]

Hi Max, Thanks for your comment! One can prove that the integral for n! converges for all complex n which have real part greater than -1. You can Google 'analytic continuation of the gamma function' for some nice articles about this. In particular, non-integer values of n! are crucial for the functional equation of the Riemann zeta function (see here: en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equation ) or 'fractional calculus', which extends the notion of the 'nth derivative' to non-integer values of n.

[@reinersim8144]

But do you have any videos on fractional calculus or any way to define transformations in porous media?

[@BPEMEHA]

So bloody good, thanks, mate.

[@utuberaj60]

Great video. Deducing the factorial from this integral using the Leibnitz Rule of Differentiating Under the Integral Sign (DIUS) is indeed a novel approach. There is only ONE problem in my understanding of the definition of the Gamma function - please pause at 7:58 min of the video. Setting A=1 you have got the needed function But initially you had DEFINED the Gamma function where "e" is raised to "nx". This does not match with the final answer you got pretty nicely at 7:58mins. Secondly,just yesterday I saw another video by BlackpenRedpendefining the "Pi" function = /int (0 to infinity) x^n *e^-x = n!. The Gamma function is defined very similarly except that n-->n-1. This is the standard definition in all text books. Can you please check and revert please?

[@alisiraydemir]

I believe it is just a typo, it should be "a" not n in the first case.

[@reinersim8144]

Great explanation! Concise yet complete!

[@MIVSIO]

Explanation is just too awesome

[@matthewjchamplin]

"1 to the power of anything is always one" *laughs in complex number*

[@Tannz0rz]

Fantastic proof, thank you.

[@34sonamrani45]

kzbin.info/www/bejne/oXLLaGyoltOFsJY

[@suzum0978]

This was gorgeous

[@Noah-jz3gt]

I like your channel name 🤣and thanks for such a neat proof!

[@Manuel-pd9kf]

Great explanation

[@Sumner18]

What are the implications from setting A to be a different value at the end instead of A=1?

[@LearnMathsFree]

Hi Sumner Losenn, No real implications as far as I know, other than that you'd get an expression for the integral of x^n e^(-tx) for suitable t.

[@ny6u]

the implication is simply that you would have to multiply your evaluated integral result by A^(n+1) every time in order to get n!

[@kuangmarial7390]

Very precise and understandable. Thank you for the nice lecture

[@teamzumali2914]

TYSM, u're a life saver

[@mathteacher2651]

excellent - made it look easy

[@xgiacomocalore6490]

Simply, clearly.. super 👏👏💪💪

[@bijoydas6044]

THANK YOU , THANK YOU SIR for this easy explanation

[@premkumarsr4021]

Lovely explanation

[@theprojects6640]

Wow what a great video !

[@lm58142]

The equation shown initially where e is raised to -nx is incorrect. It needs to be raised to -x for the equality to hold.

[@oringaapollo2833]

How do you differentiate a gama function

[@ny6u]

Very nicely done !

[@coreydavis6427]

A very educational video thank you.

[@anandjee2901]

Thanks bro I understood it well Execp for the thing that why you put A=1 Plz explain

[@mohfa1806]

you can choosa A to be any positive integer , for every value of A you will get different valid equation...taking A=1 will result in a very special equation which is the gamma function....

[@chessandmathguy]

Since it holds true for any positive value of A, then it holds true when A=1. And A^(n+1) is way easier to simplfy when A is 1 than when A is any other positive number.

[@Inspirator_AG112]

I am thinking it must be possible to do this with tetration.

[@aricwang5107]

Is there a reason as to why the integral of the exponential function helps prove the gamma function?

[@RefluxCitadelRevelations]

Super clear, super cool, thank you.

[@adam-fk2lu]

This is so beautiful oh my god

[@optotran3743]

Question: How is the Gamma function used to calculate the factorial of a decimal number when the "n" from the proof refers only to natural numbers (1,2,6,24...)?

[@siddanthvenkatesh2744]

I believe the factorial is only defined for natural numbers. So when extending to reals it you get choose what to do. As long as it satisfies rules of factorials like n! = n * (n-1)! and equals the factorial for natural numbers you should be good.

[@thefantasticterrormelon1363]

Great explanation! :D

[@spdas5942]

It was devived for positive integer and no derivetive 0. Is it true for fraction and complex number ? Kindly justify. Thanks and hat’s off for the amazing video sir ! Long live sir. ❤ U.

[@sukanya4498]

Awesome Video! Thankyou ! ❤️🙌🏼

[@digbycrankshaft7572]

Nicely done

[@o3235]

The problem with this proof is it's shown if n is integer but this also work for fractions

[@Omaryllo]

what happened to the n in the power of e?

[@yurfwendforju]

Yea but I believe this only proofes it for integers right? Why dies it now work with all Real Numbers?

[@vairavanvairavan4844]

Post the proof for the " Leibnitz integral rule".How Leibnitz just postulated this theory for differentiation under integration.

[@xgiacomocalore6490]

Very fine!

[@timandersen8030]

What's the intuition for picking that initial integral e ^ -ax and later differentiating both sides?

[@mohfa1806]

wow...salute

[@physicsperadox7849]

What is the physical significant of gamma function

[@saidmurodsultonov1184]

Thank you

[@bjdthethreecoolmathfolk2528]

Thanks this helped

[@ahmadmassalkhi598]

Amazing

[@virusalgebra33]

Everything thing is OK. But the Wikipedia (link that u provided in the description) says that interchange of differential operator and integral operator under suitable conditions. You may note that integral is definite. But in this gamma function, the integral is indefinite. Can you explain it please?

[@subhradipporel285]

if we had started with plus A , I think something strange must have happened ... it will show that sum under the curve x to the power n scaled by e to the power x converges which is absurd

[@34sonamrani45]

kzbin.info/www/bejne/oXLLaGyoltOFsJY

[@virusalgebra33]

YES

[@PULOK-CSE-21]

You so awesomely described brother🌹🌹🌹🌹😬😬😬😬god bless you🥰😍😍😍

[@bijoydas6044]

👌👍 sir, I have a doubt in infinite limit definition of gamma function. is there any proof of limit(n→∞){(1*2*3*.....*n)*n^z}/{z*(z+1)*(z+2)*.....*(z+n)}=(n-1)! or (z)

[@DHAVALPATEL-bp6hv]

BEEEE UUUUU TTTT FULLLLLL !!!!!

[@boogychan]

Beautiful 😍

[@DeepakGupta-uv1oh]

Super method 🎊

[@ikhsanmnoor8589]

Proofing much more fun doesnt?

[@abd-elrahmanmohamed9839]

Woooow

[@ferashamdan4252]

❤

[@chandranisahanone]

Euler🗿🗿🗿

[@louisromao7183]

Be careful. The integral you ended with is different than the integral you started with.

[@abdulrhmanmuhammad7398]

Wooww