Can we have negative factorial? 

OMG! You are right! I forgot for that bound is from 0 to 1... This is what happens when I have to do two different types of improper integral back to back...

Are you argentinian?

@@Prxwler ttrrrryyyrfi

What is the Integration of {x+1/x}½ ? Please solve this problem.🙏🙏🙏

You can do it like this: When doing Π(z), plug in complex integration.

💀

​@@SEBithehiper945 How to actually calculate the negative number factorial without the intervention of gamma function plot. I want to plot (-1/3)!,(-2/3)!,(-5/3)!,... etc. i tried to solve by gamma integral. But didn't ended up in answer

R. 1

@@alexwang982 Oh really? Then n! for n

@@yosefmacgruber1920 gamma function, mate and we meet again!

@@alexwang982 using gamma function (-1)! diverges, and you cant even say that you can arrange it infinitely many ways, limit of x! as x->-1 doesnt exist

also gamma of -1 is 0!, remember your definitions

You are too uneducated mathematically for this channel

Robin Sailo I think he meant -1/2 of a quarter (coin)

@@arnavanand8037 Ohh and you're too educated for a joke?

@@arnavanand8037 get over yourself buddy

@@arnavanand8037 you too sit, have a nice day.

The Π function (mentioned in the video) is what you’re looking for, it’s the Γ function, but displaced by 1 unit: Π(x) = Γ(x+1).

@@GRBtutorials thank you!

blackpenredpen so what's 0.5! ?

Yes, cause e^t in the interval [0,1] is always less than e, so if you replace e^t with e the value gets smaller. and cause e is a constant it can be ignored entirely.

Chvocht - thats how i found this video aswell, dont even know why i clicked on it

Peter Auto r/wooosh

Yes *Leans Chair Backwards*

@@PeterAuto1 I know this was 4 years ago but woooooosh dude

Amazing

Jorge Eduardo Pérez Tasso in which exam bro

In one of my College's exam @Hritik Rastogi

Nicholas Leclerc Yes

Do you live in ישראל (Israel)?

If we treat the number line from negative infinity to positive infinity as a infinite circle then the undefined part is when both ends meet at infinity, I think they showed that each undefined part has its own infinity, I think there's some mathematical theory that uses that.

Not necessarily. Remember, Infinity is a concept, so if you want, you can treat infinity like a number, but not exactly like one. Like the idea of ∞+n=∞ and 1/0≠+∞ or -∞, but instead is 1/0=±∞

@@VenTGM09 What do you mean "not necessarily", only to reply with something else that doesn't follow up on the original comment

@@orngng did you even look at the last part :| Listen, (-1)! gets you a vertical asymtope as you can literally see in the graph of Π(x), and a vertical asymtope has the value of 1/0, which could possibly be ±∞. The original comment literally described a vertical asymtope is and why it's a problem to 1/0. Do YOU understand what the comment is even saying?

@@pon1 That is called "Wheel Algebra", my friend.

It's all one. Check the date of the "wau" video.

Would ∏(n) = n • ∏(n-1) be an improvement upon your syntax, or did I do it wrong in some way?

Take the Gamma function's graph and shift to the left by 1, because Pi(x) = Gamma(x+1).

sorry.... I think I forgot to lower the volume on that..

That would be cool just try and plug it in and see what happens

@@bonkuto7679 I don't think there is a meaningful or useful notion of what it means to raise a number to a quaternion exponent power

@@wraithlordkoto maybe not in today’s conditions of math and science

@@The-Devils-Advocate I dont remember what it means, but quaternion exponents are a thing actually

@@wraithlordkoto I meant that they might not be useful today, but later they could be, like imaginary numbers

*plane. Oops.

hehehe

Like I have said many times before, It is *_undefined_* due to *_definition issues._* A common solution used by the math community is ±∞ as it's own value instead of +∞ or -∞.

Trick question. If she sails -- although _rowing_ would be more consistent with her stated mode of transportation -- faster than she runs, and the shortest path is only rowing, then obviously taking the shortest path is not only the path of least distance, but also the path of least time. Rowing the *sqrt((3^2) + (2 ^ 2)) = 5 miles* (by Pythagoras' Theorem) at *6 miles / hour* would take her 50 minutes. All other paths are slower than that. There is a much more interesting type of problem that's similar than this, but it only works if the speed in the water -- or whatever travel medium the starting point is in -- is actually _slower_ than the one on the sand (or whatever type of medium the end point is in). It also only works if the end point is _not_ on the line that is the transition from one medium to the other (the shore in this case). Instead, it must be at least marginally "land-inwards", so to speak If you're interested, watch this video by VSauce. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-skvnj67YGmw.html The whole thing is brilliant and I definitely recommend watching the whole thing, but the type of problem I was talking is given an example at around the 6:25 mark (or maybe a few seconds after that -- it's the one with the mud and the road).

Well the result is right...you still end up with infinity :)

I will try!

@@blackpenredpen And what about quaternion factorials? Is there any such thing? Are quaternions the ultimate numbers?

Yes, it is analytic continuation using that identity.

Yes, GAMMA in complex analysis is a meromorphic function, it has poles with residues at negative integers, and you can compute the integral in all the positive complex domain (Re(z)>0 otherwise the integral in undefined). Beware the real part of z being negative though since you need the mirror to compute the analytic conitnuation, example , GAMMA (-3.15) = pi/sin(pi*(-3.15_)/GAMMA(4.15), same thing goes for any complex value with negative real part, you need to mirror into the original domain, GAMMA(z) = pi/sin(pi*z)/GAMMA(1-z)

materiasacra

Yes, since Re(z)=1 >0, the integral is convergent, GAMMA(1+i) = .4980156681-.1549498284*I

Whatever the hell a "factorial" of a non-ordered field means. :-D

reddit.com

Adam Kangoroo the derivative of the GAMMA function is called the digamma function and it is also meromorphic with the same poles

Eventi with that technic it doesn't converge

Set tan(u)=2x, then sec^2(u)du=2dx. After that you end up with half of the integral of sec^3(u). You can find the solution to that on his channel. Then use the fact that u=arctan(2x).

kaszimidaczi Oh thank you! I haven't though it could be possible with tangent. Thanks :D

Simply set 2x = sh(u), then 2dx = ch(u) du, dx = ch(u)/2 du. Putting everything in your integral and knowing that 1 + sh^2(u) = ch^2(u) and ch(u) is always possible, you get ch(u) * ch(u)/2 du = 1/2 (ch(u))^2 du. After that use fact ch(u) = 1/2 (e^u + e^(-u)) and you'll get pretty simple integral with exponents.

Rafa xD No problem :)

His name is Steve?!

Its a meme you dip

Yes, but the same limit approached from the left approaches -inf, hence the "undefined".

Agreed: There's a nice plot of the Gamma function (not the Pi function) at Wikipedia: en.wikipedia.org/wiki/Gamma_function

@Gerben van Straaten Agreed: The value is undefined, because you reach different limits if you approach from left vs right. In fact, my comment shows that approaching -1 from above (i.e., x --> -1+), it approaches infinity.

Also, putting in any n+1 to the gamma function... You'll find that it gives the same integral as the pi function by subtracting 1! The gamma function has a very useful property that gamma (x+1)=x gamma (x) which flirts very closely with the Reimann Zeta Function and a ton of other series in higher math

Gamma function is usually much more convenient when studying the Riemann Zeta function

gamma function also arises in statistics very naturally.

@@tracyh5751 yep! Gamma distributions are useful for modeling continuous random variable distributions that are positively skewed. Also, other distributions like the chi squared distribution, and the exponinetial distribution are really special cases of the gamma distribution.

The gamma function is indeed very convenient for the riemann zeta function, but what i really don't get is people using it for calculating simple factorials, WHY??? You are doing more work when you could be using the capital pi function which is simpler.

wo997 we don't usually use factorial but GAMMA function. And yes you can compute GAMMA(1/3) and by Euler reflection the other ones. This number is transcendental (PROVEN!) I guess what you mean is 1/3! = GAMMA 4/3

himanshu mallick zeta integral is only convergent for Re(z)>1 the same thing for GAMMA function (Re(z)>0). For any negative complex number I mean the real part you need to use the mirror for the analytic continuation (functional equation)