Gaussian Integral 6 Gamma Function

Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out!

In this video, I calculate the Gaussian integral by using properties of the gamma function, which is a real-variable analog of the factorial function.

[SartajKhan-jgnz]

Now this is awesome. I hate to use Multivariable and moreover I dont even understand the Jacobian identity.

[Czeckie]

this is my favorite series, I hope netflix will pick it up for the season 2

[mehdisi]

Very beautiful Calculation. I am more convinced of this calculation than previous calculations of the Gaussian integral. I hope that your other solutions to the problems will be as simple as that 😉.

[Gameboygenius]

The most interesting way so far. Watching these are like riddles, where each time I try to figure out here the delicious π will come from.

[michelkhoury]

Very nice solution! I said that to evaluate gamma(1/2) let evaluate (gamma(1/2))^2 so we have: gamma(1/2)*gamma(1-1/2) and by using the formula where 0<a<1, then (gamma(1/2))^2= pi/sin(pi/2)= pi for a=1/2 so gamma(1/2)=sqrt(pi) since the gamma function is positive and gamma(1/2) by definition is int(t^(-1/2)*exp(-t)= 2* int(exp(-x^2) from 0 to infinity) by making the substitution x=sqrt(t) therefore int(exp(-x^2) from 0 to infinity) = sqrt(pi)/2 and that's my solution 😀

[jayamitra]

And could you say what's the name of the formula you used for gamma (x+y)?
That trig sub was superb

[filipbronola]

Good stuff, wish this guy was my math professor in every class, seems super passionate :)

[jamesbentonticer]

Did you do all 12 of these great videos in one day?

[D-Bar]

This is definitely the easiest and simplest way to doing it

[MrCigarro]

En la mitad del camino de esta serie, Estimado Doctor Peyam, le puedo decir que hay dos series que me mantuvieron ( y aún me mantienen) expectante: Cosmos, y esta serie. Soy un estadístico profesional, así que esta serie es súper importante.

[chandankar]

Hey peyam ! Please do a live session to hang out with your subscribers and take questions live.

[jayamitra]

Probably gonna be my favourite, cuz nothing beats gamma😀

[tgx]

This is great solution, without "polar".I'm in better mood already, at least for a while.

[typo]

Why are we taking the positive square root at the end?

[shiina_mahiru_]

speaking of the gamma function, there is something funky with that. I don't know who was the "ABSOLUTE genius" guy came up with this "genius" notation, but I would prefer I want to know his "genius" reasoning for choosing a notation that's not consistent with the integrand and the output

[alvarezjulio]

Next serie: Stirling approximation of factorial in 12 different ways.

[UrasSomer]

You don't have to "assume" things are positive. t is strictly positive on the interval 0 to 1 anyways.

[benjaminbrady]

If you were just going to pull on a gamma definition, it would've been much easier to use the reflection formula

[swordofdoom]

Another great thing would be euler reflection formula

[GreenMeansGOF]

I thought you were going to use:
Γ(1/2)*Γ(1-1/2) = π/sin(π/2)