Prove an integral representation of the zero-order Bessel function

In summary, there is an integral representation of the zero-order Bessel function in section 7.15 of the book "Laser Physics" by Milonni and Eberly. This equation can also be found in the paper "Exact solutions for nondiffracting beams. I. The scalar theory" by Durnin. After some adjustments, it can be rewritten as J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha\rho*cos(\phi'-\phi)]}d\phi. This equation is not found in some commonly referenced sources, but can be derived using the given formula and some substitutions.
  • #1
Dale12
19
1

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:

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

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 [itex]x=\rho cos{\phi}, y=\rho sin{\phi}[/itex].

The Attempt at a Solution


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

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:
[itex]J_0(t)=\frac{1}{\pi}\int^{\pi}_0 e^{itcos{\phi}}d\phi[/itex]
this integral from 0 to [itex]\pi[/itex] could be rewritten to [itex]2\pi[/itex]

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

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!
 
Last edited:
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  • #2
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!
 

1. What is an integral representation of the zero-order Bessel function?

An integral representation of the zero-order Bessel function is a mathematical expression that represents the zero-order Bessel function (J0) as an integral of a certain function. It is given by J0(x) = 1/π ∫cos(x cosθ) dθ, where x is the variable and θ is the integration variable.

2. How is an integral representation of the zero-order Bessel function derived?

An integral representation of the zero-order Bessel function can be derived using the method of Laplace's integral, contour integration, or the Fourier series expansion of the cosine function. Each method involves manipulating and simplifying the integral until it takes the form of the integral representation.

3. Why is an integral representation of the zero-order Bessel function useful?

An integral representation of the zero-order Bessel function is useful because it allows for easier computation of the Bessel function in certain cases, such as when x is a large or complex number. It also provides a deeper understanding of the properties and behavior of the Bessel function.

4. Are there other integral representations of the zero-order Bessel function?

Yes, there are several other integral representations of the zero-order Bessel function, including the Hankel transform, the Mellin transform, and the Jacobi-Anger expansion. These representations may be more useful in certain applications or for different ranges of the variable x.

5. Can the integral representation of the zero-order Bessel function be generalized to higher-order Bessel functions?

Yes, the integral representation of the zero-order Bessel function can be generalized to higher-order Bessel functions (Jn) by replacing the cosine function with the appropriate trigonometric function, such as the Bessel function of the first kind (Jn) or the Bessel function of the second kind (Yn). This results in a similar integral representation for the higher-order Bessel function.

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