What is the proof for Dirichlet's formula for integrals in atomic theory?

[andresordonez]

Hi, I'm reading a paper about integrals that occur frequently in atomic theory, and the Dirichlet formula is mentioned.

\int_0^\infty\mathrm{d}x \int_x^\infty F(x,y)\mathrm{d}y = \int_0^\infty\mathrm{d}y\int_0^yF(x,y)\mathrm{d}x

I'd like to see the proof of this formula, in what book can I look for it?

I googled it but only found a reference to it, in what might be an equivalent form (is it?).

http://mathworld.wolfram.com/DirichletsFormula.html

Most of the results of the search are related to the class number formula (I don't have idea about number theory but I guess this isn't what I'm looking for)

Doesn't seem to be a very famous formula or maybe it is known with another name. Anyway, I'd like to see where does it come from.

Thanks!
(by the way, if you find anything wrong with my english please let me know)

 

[snipez90]

This is a particular case of "[URL theorem[/URL].

This http://kr.cs.ait.ac.th/~radok/math/mat9/04c.htm" gives a proof of the desired result. I think it is from Differential and Integral Calculus Vol. II by Richard Courant, which I guess is reprinted online.

 

[bkarpuz]

Set \varphi(t):=\int_{0}^{t}\int_{x}^{t}F(x,y)dydx-\int_{0}^{t}\int_{0}^{y}F(x,y)dxdy and show by Leibniz rule (provided that F is continuous) that \varphi^{\prime}(t)\equiv0, i.e., \varphi is a constant function.
Then, complete the proof by using \varphi(t)\equiv\varphi(0)=0 for all t.