Half factorial using the gamma function

In this video, I computed half factorial using the gamma/pi function

[mikefochtman]

I like how you paused during the u-substitution and verified that the boundaries are still correct. I sometimes miss that step.

[alfsn]

i think it was actually good that you left the wrong 2 and just posted the video.
a great teacher does not require perfection!

[coreymonsta]

I love the way you teach man. I teach the same way based on intuition by figuring out what happens next on the spot. It helps the class that way

[Zerotoinfinityroad]

I love your videoz and never get bored....
Thank u prime Newton's ive learnt a lot from ya ❤

[adwz]

gamma(1/2) = sqrt(pi) is a lovely result that everyone should remember! If u remember the reflection formula, this should be easy to remember:

gamma(z)gamma(1-z) = pi/sin(pi*z), and plug in z = 1/2

==> gamma(3/2) = 1/2 gamma(1/2) = sqrt(pi)/2

Video Ideas: I think it would be cool to make a video on the gamma function product formulas, such as the reflection formula, the Euler product formula, the Gauss product formula, the Stirling series and the Weierstrass product formula (which can be used to deduce the digamma function psi(z) == d/dz ln(gamma(z)), which is related to the Harmonic numbers and can be used to explain the Euler-Mascheroni constant).

Plenty to take in! Can also do the Taylor series of the gamma function (related to above), the beta and the Riemann-zeta functions (all these functions are linked!)

[AbouTaim-Lille]

The only issue about the gamma function as an extension to the factorial is that : Acording to the analytic extension of a function which demands that the function is defined on a set that has a limit point (accumulation point), in order to have a unique analytique extension.

[allozovsky]

Great video! But, of course, this is not the only possible extension of the factorial – there are other gamma functions (or, rather, pseudogamma functions) and "factorials", defined for real and complex numbers. Say, Hadamard's gamma function or Luschny factorial, just to name a few. Of course, they may give different results for (1/2)! or even for 0! (just like Luschny factorial does).

[CatManTho]

♪ It's easy like Sunday moorning ♪♪♪

[highlyeducatedtrucker]

My man dropping the Commodores in the middle of a math problem. 😄

[rohitprakash]

so what happens if you try to find (-1)! by the same method?

[ianmyers]

Check the signs in the D/I method. + integral should be -, alternating signs

[betaorionis]

Excellent explanation!

So, if (1/2)! = SQR(PI)/2, can we conclude that, for example 4.5! would be, given that n! = n*(n-1)! in all cases:

4.5! = (4.5)*(3.5!) = (4.5)*(3.5)*(2.5!) = (4.5)*(3.5)*(2.5)*(1.5!) = =

Would that be correct?

[m.h.]

I know that nowadays the gamma (or pi) function are considered as definition of the factorial, but it seems like circular reasoning to me.
You basically created a function that outputs the factorial as a result, which is fine. But now you are using that function to calculate something, that the factorial wasn't designed for and just decide that it is now part of factorial.

To make it short, you do this:
A → B
B → A
But how do you know, that "→" can be reversed? How do you know, that B didn't add something to A that wasn't there before or removed something from A, that is now missing (in terms of functionality, not value)?

[temporarytemporary-fhdf]

Me: applause like everyone else 👏👏
Me too waiting for an extra someone trying to factorize e, and Pi ...

[DEYGAMEDU]

why sir behind 0 why these type of graph

[TheKhalamar]

I'll just keep distributing the factorial operator to both the numerator and denominator. (3/4)! = 3!/4! = 1/4. Much easier. Incorrect, but much easier.

[InverseTachyonPulse]

So, one could also say that π = (2 * (½!))² 👀

[christopherakenfelds]

I follow this well until you create the little chart with headings "D" and "I". Then I lose the logic of it. I don't know where that chart came from.

[GeorgePenchev-dz]

Try to find the minimal value of n! Where n us positive

[danmart]

It is not true that the function is continuous as originally defined. True, we can plot the factorial "function" on a graph, but it will have only discrete natural numbers. Then we can draw lines between the points to pretend that the function, magically, becomes "continuous." Let's be accurate about how we should describe this attempt to find factorial values that "lie" between integers: "IF we ALL AGREE TO PRETEND this function were to be continuous by some miracle, then we calculate that the factorial of, SAY 3.5 IS THIS OR THAT. But in reality this does not exist because the definition of factorial is the multiplication of adjacent INTEGERS, not partials. Of course, mathematics being what it is, we can always reinvent the definition to fit our unnatural known as "mathematicians with too much time on their hands!""