Introduction to the Gamma function & the Pi function (extending the factorial!)

The usual definition for a factorial only works for positive whole numbers, but how can we take the factorial of any number? Here we will discuss the Pi Function which is defined in terms of an improper integral and it is also the cousin of the Gamma function. I also show the Pi function's properties, which helps extend the factorial. As a bonus, I also show why zero factorial is equal to 1.

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[UB]

Fermat, on proving his Last Theorem: 17:30

[Hello-fbsp]

"-Hey mate can you tell me what's the factorial of 1?
-Yeah sure *pulls out integration formulas and l'Hopital's rule*"

[hattrick]

It is incredible what can be done with Euler's number. As time goes by you really begin to appreciate that number more than older societies appreciated pi

[josealbertolopezlopez]

Why is Gamma more popular and used than Pi? Pi seems more logic if you want a function that extends factorial.

[derendohoda]

Around 12:00 you're discussing using L'H n times to kill the term but the whole point of this exercise is to create a function when n isn't an integer. If n isn't an integer in this step, you can't apply L'H n times to get a factorial like you say. Really you're getting n*(n-1)*... until the t term moves to the denominator, then you get a constant divided by infinity which does have the limit 0. Minor technical point. I love your love for math please never stop!

[ZipplyZane]

It seems to me that you don't need the gamma or Pi functions to show that 0! = 1. You just need the two definitions you gave.

1! = 1
n! = n * (n-1)!

Plug in 1 for n, you get
1! = 1 * (1-1)!
1 = 1 * 0!
thus *1 = 0!*

[aaronbs]

Teacher: "Can you find a function so that f(1)=1 and..."


"a million brain cells pops up at once on your head"

[brycelunceford]

"That box means a lot to us" 😂 I died.

[hamez]

I am so happy you are doing this! Ive looked online for a reasonable way to understand the factorial function outside of just positive integers and have found nothing so far except this!

[scottjames]

If you want more, here is the wikipedia page. Wise words

[anjanmukherjee]

pure mathematics is the most beautiful subject according to me

[mrnogot]

You are the best math channel on YouTube. 3blue1brown is great and all but you get much more into the nitty gritty. Thanks man.

[chimetimepaprika]

Dude, you're such a good teacher! I never fully got why this worked until now.

[lightningblade]

I never laughed so hard while watching a mathematics video on YouTube - 16:06. Thank you so much for the video man, I've been trying to understand the Gamma function for so long and your video explained it flawlessly.

[Zonnymaka]

Euler again of course :)
Usually these kind of function are "deducted" by reasoning about "what you want" (as RedPen stated clearly in the video) and "which function is more suitable to fulfill the requirements".
Usually e^x comes up everywhere because of his extraordinary properties.
Well done RedPen!

[skoockum]

This is fantastic. I am so glad I found this channel. Kelsey's videos and Mathologer are terrific, but the best way to explain math is to walk through it step by step on the board. I've looked at the gamma and pi functions on Wikipedia and the bit with x and t in the integrals had me stymied, but here at the end of your video when I looked back at the first integration I had none of my earlier confusion-- the x's role was immediately obvious, and I never even thought about it during the entire video. Looking at a page full of calculations it takes a lot of work to decode the operations and relationships. But watching it unfold in front of you is a cakewalk. LOL It's the next best thing to homework.

[sarojpandeya]

You are one of the great youtubers.
And very good maths teacher I like.

[nrpbrown]

This was juat the explanation for
both these functions ive been looking for, thank you.

[Patapom]

Great!
How Gauss did come up with this anyway? And why is the gamma using x-1? Why not using the PI function directly?

[injanju]

Finally! But how dd they come up with the Pi and Gamma functions?