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 :
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.
By defintion satisfies
Now here comes the cool part.
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.
so ( since )
The integral we want is