Yes , AGM is one of the most famous identities , I have a proof of it but of course it is not due to me , I was inspired to give a complete proof by this problem from American Mathematical Monthly (AMM 6672 ) :
Show that . Then use this result to show that :
Oh , I have made a very big mistake on the first problem , sorry about that
The integrand should be squared :
Not having heard about Bessel function , I thought I would not be able to solve this until I found this formula on a website :
Therefore , your integral
My method is to switch the order of the integral , followed by substitution , then switch back its order :
Sub. we have
Then switching back !
Integration by parts ,
Changing the order of integration solely for the purpose of making a substitution? Where you do come up with these ideas?
The solution I had in mind doesn't use the integral representation of the Bessel function.
let
then using
where and
By defintion satisfies
Now here comes the cool part.
let
and
This directly implies that .
Then using the facts that and , you'll find that and
Solve the system of equations simultaneously to get homogenous second order linear ODE which has the general solution
must be zero because is a decreasing function, and must be because Bessel functions are normalized.
but
so ( since )
The integral we want is