Prove an integral representation of the zero-order Bessel function

[Dale12]

Homework Statement

In section 7.15 of this book: Milonni, P. W. and J. H. Eberly (2010). Laser Physics.
there is an equation (7.15.9) which is an integral representation of the zero-order Bessel function:

J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha(xcos{\phi}+ysin{\phi})]}d\phi

This equation could also be found in this paper:
Durnin, J. (1987). "Exact solutions for nondiffracting beams. I. The scalar theory." Journal of the Optical Society of America A 4(4): 651.

Homework Equations

here x=\rho cos{\phi}, y=\rho sin{\phi}.

The Attempt at a Solution

Rerwite it as:
J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha(\rho cos^2{\phi}+\rho sin^2{\phi})]}d\phi
this lead to:
J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha\rho]}d\phi
and after integral of \phi, this becomes
J_0(\alpha\rho)=e^{i[\alpha\rho]}?

Also I tried to look it up in the handbook of mathematics by Abramowitz, M. but failed to find this equation, except one like this:
J_0(t)=\frac{1}{\pi}\int^{\pi}_0 e^{itcos{\phi}}d\phi
this integral from 0 to \pi could be rewritten to 2\pi

J_0(t)=\frac{1}{2\pi}\int^{2\pi}_0 e^{-itcos{\phi}}d\phi

as http://math.stackexchange.com/quest...ic-integral-int-02-pi-e-2-pi-i-lambda-cost-dt
describes.

yet, this is not what I want.

Still, this equation is not found in some wiki pages:
http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
http://en.wikipedia.org/wiki/Bessel_function

Thanks for any reply!

 

[Dale12]

Well, I have found that the error appears in step 2.
It should be phi' instead of phi.
after that, it should be rho*cos(phi'-phi) above exp.
and then the integral would be J0.

Thanks anyway!