what is half factorial? (1/2)!=? via. Gamma function

[@blackpenredpen]

Gaussian Integral, kzbin.info/www/bejne/qnSzgoall9Z5npYsi=Ywmf23oz3vyQMCng

[@DarkMage2k]

"Guys?! How many ways can I arrange half of this apple?!" "Oh that's easy √π/2 ways" "Wut" "Wut"

[@martingaggero8462]

"Oh, that makes sense, bro."

[@NeerajSingh-or9hd]

easy question

[@RazorM97]

i don't get it

[@bizzybuzz2198]

@@RazorM97 a number of ways you can arrange N objects is N!.

[@RazorM97]

@@bizzybuzz2198 so true, but i think that's more well limited, factorials are limited, idk why.. it's my opinion. thank you tough

[@ffggddss]

IMO, a nicer/prettier/cooler result is (-½)! = ∏(-½) = Γ(½) = √π Regardless, thanks for another fun math excursion!

[@blackpenredpen]

ffggddss I actually did that too!! We do think alike!

[@Barteks2x]

One is an obvious result of the other: √π / 2 = (-1/2 + 1)*√π

[@EngSeifHabashy]

Well that's nice but it should be sqrt( PI) /2

[@sttlok]

@@EngSeifHabashy no, it's the gamma of a half, not a half factorial.

[@mangalvnam2010]

So, that's technically a FRACTORIAL, right? lol

[@blackpenredpen]

YES!!!!

[@arnavanand8037]

Fractoreo

[@bamberghh1691]

@@arnavanand8037 factorio?

[@happypiano4810]

Nice.

[@pepebriguglio6125]

Nah ... more like a fictorial 😛

[@machobunny1]

Been a long time since I was taught most of this stuff, but never by a teacher who could pull together so many facts to make the job so much easier. Students are lucky today...can watch it over and over, and practice with the tutor right in front of you all day.

[@guitarraccoon1541]

You're like the Bob Ross of math.

[@bonbonpony]

Except with less of happy little accidents ;)

[@MisterPenguin42]

"Happy little decision trees"

[@MrArtbyart]

Bob Ross was a glorified less than mediocre artist.

[@smvwees]

That honour goes to Tibees, though combine it with the intellect of Blackpenredpen.

[@zachansen8293]

@@MrArtbyart You're confused if you think it was his paintings that were his art.

[@ramez2775]

Someone has been eating too many factOREOS.

[@rasmussuonio3014]

Fact Oreos So funny

[@jackdulin8242]

If watching one of your videos means eating one factOREO, then I would be as fat as a house.

[@Xnoob545]

facthydrox

[@jyl123]

ur actually so funny mr. integration bee winner

[@Mohamed-Taha-Lakhnig]

I study in morocco And my teacher told us to search about it . This video is awesome . I watched before 3 ones but I didn't understood . Now It's more clear for me Thank You

[@MG-hi9sh]

Wow, this was very informative and interesting. I love the depth and variety of the topics covered on this channel. Good work man. You are doing a very good job.

[@wanyinleung912]

3:46 You will also be infinity

[@convergentradius]

Thanks for boosting my confidence.

[@hamster8706]

Thanks

[@NBx03]

"Is it t^2?" *"no u"*

[@radiotv624]

Very interesting, I love videos that broaden somewhat narrow definitions. I’ll keep this one in my back pocket!

[@avtaras]

Much better than most tutorials. Thanks for being clear!

[@AlejandroGomez-yx1sg]

Beautiful! Congrants and thanks for all your teaching efforts. Greetings from Colombia 🇨🇴

[@MartinPuskin]

I would love if you showed how to find the local minimum of the pi function (between 0 and 1)

[@sbok9481]

From calculus we have learned that at a minimum point (or maximum), the gradient=0. In other words, the first derivative is 0. So go ahead and take the derivative of the pi function and put that = 0. Solve for x, then you will get it.

[@Jacob-uy8ox]

One of your best videos! Also de Pi function and the Gamma function!!!

[@mathonthego1947]

I'm new to this channel, and I'm pretty much hooked after seeing 3 videos.

[@blackpenredpen]

Math On The Go thank you so much!

[@brandongammon6978]

Bro u got some fire clothes

[@blackpenredpen]

Thanks

[@brandongammon6978]

@@blackpenredpen very smooth, keep doin u

[@gongasvf]

I just remembered what love feels like

[@blackpenredpen]

Gonçalo Ferreira yay!!!

[@spiritgoldmember7528]

Can the Pi function take negative values for x?

[@blackpenredpen]

Yes, (-1/2)! is sqrt(pi)

[@materiasacra]

The integral representation shown in the video is well-defined for x > -1. Below x = -1 the integral no longer converges at the lower bound t=0. It is possible to extend the definition of Pi(x) by analytic continuation. This yields a unique value everywhere in the complex plane, _except at the negative integers_ . So Pi(-3/2) exists, and comes out as -2 sqrt(pi).

[@materiasacra]

@densch123: But it will diverge at t=0.

[@wurttmapper2200]

Not negative integers

[@jimallysonnevado3973]

yes except for negative integer

[@muhammadmubbushirhussain4734]

I really like your videos U made my calculus easier LOVE FROM PAKISTAN 🇵🇰

[@fantiscious]

4:12 BPRP: "2udu" My brain: "To you the you"

[@Asdun77]

you did a great explanation allah bless you

[@potatopassingby]

what if we put a complex number in the Π function?

[@MattMcIrvin]

It's defined on the whole complex plane, except for the negative integers. Usually what you see plotted is Gamma(z), which is Pi(z-1): en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma1.png en.wikipedia.org/wiki/Gamma_function#/media/File:GammaAbsSmallPlot.svg

[@sasidharankarthikeyan3798]

It has been 10 years since I derived this equation in my graduation. Feeling nostalgic.

[@gusthomas6872]

Are we allowed to call you 老师?

[@blackpenredpen]

Gus Thomas I am one already. So yea!

[@gusthomas6872]

blackpenredpen 谢谢老师!

[@AlgyCuber]

are you a math teacher?

[@XdarkmutantX1]

laoshi

[@blackpenredpen]

Statiscube It means " teacher "

[@uumlau]

This reminds me of the whole 1+2+3+4+...=-1/12 bit. Why? Because "factorial" has a specific meaning (N! = N*(N-1)*(N-2)*...*1) that doesn't apply for non-integers. So the question of "(1/2)! is equal to what?" is a meaningless question, just as "What is the sum of all positive integers?" is a meaningless question. In both cases, something subtle is going on: the question being asked has slightly changed! In the (1/2)! case, the question has become "How would factorials work for fractions, while remaining self-consistent?". In the sum of all positive integers, the question has become, "My math has me adding all positive integers, but I know my final answer is finite? How can we self-consistently evaluate this to get a finite result?" Without the subtle change in the question being asked, the results don't make sense. (1/2)! isn't the combinatorial result of anything. And -1/12 isn't the SUM of all positive integers. In both cases, the math is being extended to handle "what if?" scenarios that in turn help us solve problems quite different from the contexts of the original questions.

[@mandeltownthekillerfrombab5202]

it's a "Numeral Theorem Superstition"

[@DasZeppeli]

Things like that are why I love maths ❤

[@LudwigvanBeethoven2]

If (1/2)! = sqrt(Pi)/2 Then any real number with 0.5 as fraction will have PI in it's factorial. Thats cool

[@rj-nj3uk]

I am making notes of whatever you teach for my semester exam.

[@DjVortex-w]

Why do pi and e appear everywhere?

[@blackpenredpen]

WarpRulez that's why they are famous!!

[@georgepennington908]

Because they’re magical numbers

[@terracottapie]

But 0, 1 and 2 are labels we assign to very common core concepts that really do appear everywhere. Pi isn't the same as 0, 1 and 2. It's a crazy, esoteric, infinite non-repeating decimal. It's a name we give to the ratio between the circumference of a circle and its diameter, and yet it appears in all sorts of stuff unrelated to circles. It's pretty interesting.

[@terracottapie]

Except that one and two are easy things to conceptualize, for anyone over the age of 3, and "pi" is an abstraction that only makes sense to most people in the context of circles. So for pi to appear in random other places is fascinating to people including mathematicians. There is a cottage industry of mathematicians giving talks about how cool it is that pi appears in infinite series and physics and electromagnetism and 100 other things. Carl Sagan wrote a book where aliens solved pi to enough places that they eventually got to a coded message from God, to play off this fascination. So this isn't a concept that just appeared today in a KZbin comment.

[@bonbonpony]

+WarpRulez: `π` appears in many places where circles and cycles are involved, and this is pretty much everywhere in Physics, because matter has wave-like nature and its behaviour repeats periodically (that is, it "goes in circles"). And `e` appears everywhere because it is the "natural rate of growth" - the one in which something grows continuously, and the rate of that growth is equal to the current value of the function. This is also a very common scenario in Physics, because things tend to grow proportionally to the current amount of stuff. E.g. heat flows out the faster the more hot something is. Radioactive atoms decay faster the more atoms are there, because the number of interactions between them is greater. Same goes with economy where the more money you invested, the more interest you get each month. Also note that `π` and `e` are actually related - they're like two different ways of looking at the same thing, or two complementary mechanisms. Because you can think of turning in circles as exponentiation (which gets more obvious once you learn about complex numbers and their powers). So if `π` measures the length of the cycle (actually, a half-cycle; look up "tau" ;) ), then `e` measures the rate of change that makes this twisting possible.

[@beatriceleeknowles5944]

Sad that you didn't do the gaussian integral from first principles. It's a really nice proof which doesn't take very long and makes it clear where the square root comes from.

[@benarcher372]

I wonder what the graph looks like; pi(n); 0

[@JUGNUMEHROTRANEETASPIRANT]

You could have just sad that n!=Gamma{n+1}=nGamma{n} & Gamma{1/2}=√π. {GAUSSIAN INTEGRAL} & THEREFORE : [0.5]!=√π/2 ~ 0.886

[@raiturner2132]

My mans got style!!

[@banderfargoyl]

It's just a semantic issue I know, but saying the Pi function generalizes the factorial to non integer values seems a little different from saying the factorial itself can be applied to fractions.

[@bonbonpony]

Probably. But we could say the same about exponents in general: Originally they were defined only for positive integers. But (thankfully) the definition has been extended to zeroth exponent, negative exponents, fractional exponents, real exponents, and even complex exponents along the way. It wouldn't be so convenient if we had to use all sorts of different "special functions" to do exponentiation with whatever other than the natural numbers, would it? :q So I think the same should be applied to factorials: once they've been generalized, I see no point in using some fancy "special functions" to calculate the (generalized) factorial of `1/2`.

[@angelmendez-rivera351]

Bon Bon Bingo.

[@b43xoit]

Wolfram Alpha gives .9 -> .962, .8 -> .931, .7 -> .909, .6 -> .894, .5 -> .886, .4 -> .887 . So maybe the minimum is found close to .5 as the input. .51 -> .886592; .50 -> .886227; .49 -> .885945 so the minimum occurs below 1/2 somewhere, but above zero. .40 -> .887264 so there is at least a local minimum somewhere between .4 and .5

[@purushotamgarg8453]

At 0:13 "Whole No." are basically POSITIVE except 0. But We can do 0! in the usual way so you should just say "Whole No.".....

[@omp199]

No, because there are also negative whole numbers. He should say "non-negative whole numbers", "non-negative integers", or "natural numbers", taking the set of natural numbers as {0, 1, 2, 3, ...}.

[@avibents1253]

does it work only for half or does it work for any 0

[@Pinjesz]

Could you make a video about one fourth factorial?

[@joshuamitchell5530]

Rocking that supreme jacket love that.

[@michaeldang8189]

It would be nice to quickly go over the x-y graph of y=x!, where 0

[@mandeltownthekillerfrombab5202]

thank U so much to learn it. (I often get false result with the integral 0 to infinity)

[@ptitemoi]

Hello! Could you please do the same for some other numbers? Like 1/3, 1/5, 1/7 etc. Thank you in advance!

[@eduardgiovannyariasrincon6635]

If both of the functions lead you to the answer, what would be the real difference between them?

[@UnOrdelyConduct]

Is there any correlation to the pi/gamma function and the laplace transform? By inspection i see that the integrals are pretty similar

[@ryzeap5836]

When you haven't seen integrals but you love this

[@gian2kk]

8:03 editing wizardry

[@sodik0305]

I very like this 😁😁😁

[@PedroXpertGames]

You could also do integrations by part firtst and then use U-sub

[@Beeboysquared]

That flex jacket tho

[@aweebthatlovesmath4220]

I love factorials!!!

[@hardikkadd5114]

If we find -1! By this then does it shows t > 0, t < 0 so t € (-∞,∞)- {0} Which says that -1! = -πi 1/0 = -πi

[@aweebthatlovesmath4220]

Lol

[@lakshsinghal9570]

wooahhh!!! I understood nothing, 'cause I have to first study the pi and gamma function but it still is AMAZING..!!!

[@natehoffmaster6726]

5:31-5:36 is why I prefer differentiation over integration. Too much guessing in integration.

[@paulg444]

great video, maybe a slow review of integration by parts.

[@Tcrrxzz]

Please @blackpenredpen teach me the basics of integrals? I have never learnt what are they

[@Pokemon00158]

I love how this guy always pins the racist comments about his pronounciation lol

[@015Fede]

Yolo Swaggins they're not racist. It's an internal meme

[@mike4ty4]

That keeps those racists shamed as they should be.

[@angelmendez-rivera351]

Federico Aguilera It is racist, it is not a meme for most people.

[@AngelRivera-mc8zc]

Angel Mendez-Rivera hey me

[@JanKowalski-zz8ef]

*racism intensifies*

[@francesco5201]

you can use strirling aproximation

[@userBBB]

√π/2≈0.88622692545 0.88622692545 way to arrange 0.5 thing and you have 1 way to arrange 0 or 1 thing makes a lot of sense now...

[@GottfriedLeibnizYT]

Fascinating! But can we intuitively make a sense out of this result?

[@jeremymoss7234]

This is a funny looking ood, but he sure is good at math

[@josuehazaelmurodiaz7736]

Hey blackpenredpen, Is possible to calculate Π(a/b)? To calculate any factorial cocient. Hello from Zacatecas México

[@sergioh5515]

josue hazael muro diaz very good question. I'm sure it is

[@josuehazaelmurodiaz7736]

It's pretty much the same, you end up with : ∞ ∫ e^(u^(b/a)) du 0 And then, I have no idea jaja

[@materiasacra]

You may want to check that u-substitution... Even when corrected, it leads nowhere interesting. The Pi and Gamma functions are easy to compute only for integer and half-integer arguments. en.wikipedia.org/wiki/Particular_values_of_the_gamma_function For other values we turn to numerical methods.

[@KalikiDoom]

0:56 "for the Gamma function fans out there": Thank you for noticing the Gamma function fans!

[@jamesmorrison7989]

If pi is in the answer a circle is involved somewhere. Im sure the complex plane but it would be cool to figure out exactly how its in play here.

[@michaelbaum6796]

Great explanation as always - thanks👍

[@andyct1982]

What is the minimum value of the function Gamma defined on the positive real axis?

[@carultch]

There is no answer in terms of elementary functions and constants. The minimum occurs at x=1.461632145, with a corresponding Gamma(x) = 0.885603194.

[@tetris450]

This guy is smart

[@mobius2827]

boy got on supreme

[@davidrheault7896]

I would say GAMMA function is more popular because it came first historically (Daniel Bernoulli conjectured, Euler solved, 1720's), while the PI function came from Gauss (1830's ?), but Gauss modified Euler's product to give a complex-valued function Gauss forged monstrous theorem without giving proofs but they worked, I am referring here to GAMMA (1/3) which is transcendental (proven by Chudnovsky in 1984). Gauss computed it through hypergeometric series I have a preference for (-1/2)! =GAMMA(1/2)= sqrt(pi) = Euler's mirror with z = 1/2

[@paulfaigl8329]

Very cool guy + cool maths!

[@mike4ty4]

A remark: Gamma(1/3), Gamma(1/4), etc. - which one might be naturally inclined to consider after seeing this - do not seem to have a simple representation in more elementary terms. They just have to be taken as they are. However, there is no proof that they cannot be so represented. The theory of how to prove what does and does not have a certain kind of representation is something very interesting and it would be fun to see at least a bit of video on it (e.g. Galois Theory and the inexpressibility of the solution of a quintic or higher by radical operations in the general case, which actually appears to imply a bit further that it is also not expressible with any elementary operations at all including exp, log, and trig (the last comes from the first in the complex plane so may be a bit redundant to mention.).).

[@akinextreme8136]

Plplplplplplpllpl

[@brendonreidvictor3491]

I love the video as I have always wanted to learn more about the pi function but I am wondering if it is possible to differentiate the pi function n find some stationary points?

[@angelmendez-rivera351]

Brendon Victor You can differentiate Π(χ) for integer arguments and obtain a closed-form formula.

[@MatthewBaileyBeAfraid]

I love your videos (pity that I don’t play the "Like" game on KZbin ... or FB, or Twitter). But I did subscribe.

[@Theagchm]

What(5.07)?! To integrate the u^2, use integration by parts twice! You do maths by guess and check at a high level?....and even after all that you still had to do a sub! What did you gain by NOT doing integration by parts twice - it is really straight forward.

[@BillSmithPerson]

That crazy pi again

[@Qermaq]

So I guess Π vs Γ is kinda like π vs τ.

[@ishaan1]

Can you please do a video on the Wallis Fomula? For the integration of powers of sine and cosine in product. The one which has use of Gamma function or Pi function?

[@Harlequin314159]

Question: I was trying to find the min of the Pi function between Pi(0) and Pi(1). At first I assumed the min would occur at Pi(1/2) but was surprised to find that I think I have convinced myself the min is at Pi(1/(pi-1)). Or 1 divided by 2.14159... Neat but I am not sure I proved it fully... Any thoughts?

[@materiasacra]

How did you arrive at 1/(pi-1)=0.46694...? It is quite close to the actual value 0.46163214496836..., but off by a percent. As far as I know there is no closed form expression for this number. I guess you have made some sort of approximation?

[@Harlequin314159]

Yeah looking back i just made a bad error and it ended up being really close. I basically started with the observation that gamma(1/2)^2 was pi, and that Pi(n) = Gamma(n+1). And then I got sloppy...

[@Taterzz]

"u in red and u in black" he said it! roll credits!

[@mihaiciorobitca5287]

make an other vid about gaussian integral,right ?

[@blackpenredpen]

They are in the description box.

[@mihaiciorobitca5287]

blackpenredpen yeh,from bprp to dr. peyam's show

[@arnavanand8037]

@@blackpenredpen the video is private now?

[@Craznar]

Programmer : 3!=10, Maths : 3!=6

[@kuhlde1337]

I don't get it.

[@reetasingh1679]

Justin Johnson != is the 'not equal to' symbol used in programming languages

[@kuhlde1337]

I see. Then 3!=6 would still be true for programmers as well.

[@Craznar]

Indeed... :)

[@回归常识-u1e]

programer：3!=6 math:3!=6

[@copperfield42]

mmm and how can I make a graphic of this thing? I have to solve infinitely many integral?

[@danielvidal7163]

Integer factorials like 5! for example can be written as 5x4x3x2x1 so how would you write 1/2! in the same form?

[@marble17]

Elementary school guy here

[@the1exnay]

I didn't understand the vast majority of what you did, i guess i need to brush up on my calculus

[@wenhanzhou5826]

It went wild quite fast...

[@ixian98]

is the Gaussian Integral video not available any more? :(

[@Billy_98]

Awesome!

[@blackpenredpen]

Kaneki Ken thank you!!!

[@b43xoit]

I told Wolfram Alpha, "plot definite integral from 0 to infinity of t^x e^(-t) dt, x=0 to 1". But it didn't produce a plot, only a curve length.

[@MultiJoan09]

soooo cool! been wondering a while if there was a continuos function for n! btw how can you know π is continuous?

[@darkangel2347]

The factorial function has it only positive local minimum of X=0.461632145…. with its factorial being 0.885603… which is not far from (0.5)!

[@paulthompson9668]

Can you do a video on fractional derivatives?

[@sophieward7225]

Could you make a video evaluating gamma of i? I've been trying to figure it out all day, and I can't!

[@Rtong98]

Wow! I’ve never seen u * u * e^-u^2 being used before! Thank you

[@MrLuizSinho]

Could you do a fractional calculus video? Pleaseeeee.

[@blackpenredpen]

MrLuizSinho Peyam did already. Check it out!

[@joluju2375]

Please, teach us how to derivate Pi, and find where the minimum is !

[@fizixx]

Would you do the "brachistohrone" problem where the minimal distance is between 2 points under the influence of gravity?

[@sirzatsayn813]

If I am not wrong, factorial is for whole numbers and appears in combination formulas. What is the meaning of this?

[@kkn5523]

I suppose this just shows that factorials make sense only when dealing with positive integers and 0 and why we don't extend it's domain. Still, I may be wrong

[@sirzatsayn813]

Factorial operator, by definition is the multiplication of integers only. There are infinite amount of floating point numbers including irrational ones between two integers. Multiplication of these numbers converges to 0 between 0 and 1, diverges to infinity after 1. So I am confused. Who needs the factorial of 63.748 for example? I am not a mathematician by the way.

[@celestial_crash0]

so it should be blackpenbluepenredpen?

[@blackpenredpen]

Boonda P hehehehe