Right, it's not that much harder.

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- Dec 30th 2010, 10:52 PMDrexel28
- Dec 31st 2010, 09:53 AMRandom Variable
I've come across the following identity multiple times, but I've never been able to prove it.

Show - Dec 31st 2010, 10:33 AMDrexel28
- Dec 31st 2010, 10:52 AMRandom Variable
- Dec 31st 2010, 11:06 AMDrexel28
- Dec 31st 2010, 12:49 PMDrexel28
- Dec 31st 2010, 08:16 PMsimplependulum
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 :

- Jan 1st 2011, 07:35 AMRandom Variable
Show that

where is the first order Bessel function of the first kind.

The proof is fairly long, but simple and pretty cool at the same time. - Jan 2nd 2011, 12:34 AMsimplependulum
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 ,

- Jan 2nd 2011, 04:02 AMRandom Variable
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 - Jan 3rd 2011, 02:46 PMRandom Variable
So that this thread remains active, try the following fairly easy problems:

- Jan 3rd 2011, 03:37 PMchiph588@
- Jan 3rd 2011, 04:37 PMDrexel28
- Jan 3rd 2011, 04:41 PMchiph588@
- Jan 3rd 2011, 06:29 PMRandom Variable