gamma and e^(-x^2)
You know, I was just playing around and noticed something. I am sure you all
have thought of this before, but I just 'discovered this'. I always integrated
by the polar method.
But if we use gamma it is easy.
we can use the definition:
Since is even, we get
I know, I know, it's obvious to you all and older than Moses. But I had not
thought about using gamma before for this famous integral and it just
dawned on me. I always just used the old polar thing and never worried
about another method. Stuck in that rut, I reckon. But, if there are those
out there who like this, here it is.
How'd ya prove this little fact? :D
Originally Posted by galactus
You want me to prove that?. That is a famous gamma identity, so I did not bother proving every aspect.
But, if you would like to see it, let me get back. Okey-doke. I always just used it and took it for granted and never bothered proving it.
But, I am thinking we can play on the old polar thing to show it.
To be lazy, I would just use and let n=1/2.(Wink)
I know, you want something more tangible.
Let's see, we could start with
Then, sub in .
I will finish later. Got to go for a second.
Well, Kriz, it appears I ended up using the polar thing to do that. But I think you knew that, didn't you?.(Wink)
Now, we use the polar thing. So, I learned myself something with a little prodding from the Krizmeister. Cheers.
I suppose that infernal polar was hidden there afterall.(Punch)
I can not believe I had never bothered doing these before. Oh well, I derived something I should have a long time ago. Especially, since I like the Gamma function so much.
Ohhh, I'm sorry, I misread the question, it's actually calculating again the gaussian integral.
Cheers and Happy New Year!!!!
Since we're on Gamma, derive this one:
This is the duplication formula for Gamma.
Try using the identity and put
Happy New Years
Don't worry, I'm also a good friend of Gamma function.
But it's actually a bit interesting in finding a closed form for
You mean this one:
Well, Kriz, let' see. I have been working on it while I was waiting on your reponnse and gong to the bathroom.
It is getting late, Kriz. I am going to skip some major steps and join you tomorrow. Gotta go. But I have this so far. Need to explain some steps further if need be.
We need to show that which I think can be gotten from
Therefore, we get
Plugging this in the above gives:
Hope you can follow that. I will touch it up tomorrow.
It's getting late and my brain is beginning to see p's and q's everywhere.
The method I really love for calculating this integral is considering the function :