Hi
I am trying to learn the proof for the value of the Gaussian Integral posted here: Gaussian integral - Wikipedia, the free encyclopedia
under 'By Cartesian Coordinates"
I understand the basics of double integration, but I don't quite understand this proof.
I see that the function is an even function...but what is the significance of saying "Let x=ys and dy=xds"
I also cannot see how
arises when
is squared.
And...I also do not understand how the 's' variable arises later on into the proof.
I would greatly appreciate if these questions were answered and it would be extremely appreciated if a quick run through of this proof was posted.
Thank you very much.
Thank you!
I appreciate the thorough proof. However, this is the proof that involves polar coordination. I initially wanted an explanation for the alternative proof that involves what is posted here:
Gaussian integral - Wikipedia, the free encyclopedia
Under 'By Cartesian Coordinates"
And my main question here for this specific proof is what the significance of letting
"y=xs
dy=xds" has.
Along with this, I don't quite understand how the 's' arises in the proof.
However, I extremely appreciate your efforts in providing the proof for the Gaussian integral.
This is no different then when I let y=x. As long as we clearly define our change of variables, and change our bounds accordingly, the integral/limit is still valid.
So, in this case
were selected out of convenience. It then becomes clear that
Which becomes our exponent. Of course you can then integrate the function quite easily because we have an x out front. They compute the integral directly, but you could most definately do By Parts here and get the same result.
Once this simplifies it comes to an integral of known form and is evaluated at that point.
I am starting to understand this proof. I can see why these variables are set and how they are replaced later on.
I just have one last small question.
When given:
How does this turn into ?
Wouldn't it turn into ?
I see how it is necessary for when it's substituted later on, but...I don't quite see why x remains unchanged.
Again, thank you very much. From your previous post I pretty much understand the proof.
Note that x does not depend on s and is therefore treated like a constant when we integrate our function with respect to s. So we then have
With x being a constant and the derivative of s being one we have
Finally
I would like to note that you thought it might equal
This assumes that x has a dependence on S, which it does not (we've only defined y in terms of s). But supposing that it did, say X WAS dependent on S. The above would still not be correct, we would need to use the product rule here
Clearly this is
Which is not eqaul to what you had.
Thanks! I now completely understand that y is dependent on s, and so x is treated like a constant.
I actually forgot to ask this question in my previous post, and I think this is my very last question, anyway the question is:
So we have that
is equal to:
and I understand why this works since
But it states
So wouldn't ?
Thank you very much with the details on the proof. I'm really understanding this proof, I just need this one last clarification.