Integral of square of Bessel function

In summary, the integral of square of Bessel function is the area under the curve of the square of the Bessel function, denoted as ∫ |J<sub>ν</sub>(x)|<sup>2</sup> dx, with applications in mathematics and physics. It is significant in solving differential equations and representing physical quantities, and can be calculated using numerical methods or expressed in terms of other special functions. The integral has properties such as orthogonality and recurrence relations, and has practical applications in signal processing, electromagnetics, and quantum mechanics.
  • #1
vietha
4
0
Hi there,

I am starting with the Bessel functions and have some problems with it. I am getting stuck with this equation. I could not find this kind of integral in the handbooks.

1. [tex]\int_0^aJ_0^2(bx)dx[/tex]


Besides of this, I have other equations in similar form but I think this integral is the key to solve others:

2. [tex]\int_0^\infty J_0^2(bx)e^{-\frac{x}{c}}dx[/tex]

3. [tex]\int_0^\infty J_0^2(bx)e^{-\frac{x^2}{c}}dx[/tex]

3. [tex]\int_0^\infty J_0^2(bx)\frac{x}{c}dx[/tex]


Please help me. It is highly appriciated.
 
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  • #2
I asked Maple, and got something in terms of the Struve H function.

[tex]\int _{0}^{a}\! \left( {{\rm J}_0\left(bx\right)} \right) ^{2}{dx}=-
a \left( -2\,{{\rm J}_0\left(ba\right)}+\pi \,
{{\rm J}_0\left(ba\right)}{\rm H}_1 \left(ba \right) -\pi \,
{{\rm J}_1\left(ba\right)}{\rm H}_0 \left(ba \right)
\right) /2
[/tex]

added: This is wrong. I for got the square. This is only [itex]\int _{0}^{a}\! {{\rm J}_0\left(bx\right)} {dx}[/itex]
 
Last edited:
  • #3
Hi g_edgar,

Thank you for your reply. I tried with Maple too and I got this:

[tex]a*hypergeom([1/2, 1/2], [1, 1, 3/2], -a^2*b^2)[/tex]


The equation you got must be the result of this integral: [tex]\int _{0}^{a}\! {{\rm J}_0\left(bx\right)}{dx}[/tex]


I have to search for the generalized hypergeometric function. I have a little knowledge on this.
 
  • #4
I looked these up in the book "integrals of bessel functions" by Luke, McGraw-Hill 1962.

vietha said:
Hi there,

I am starting with the Bessel functions and have some problems with it. I am getting stuck with this equation. I could not find this kind of integral in the handbooks.

1. [tex]\int_0^aJ_0^2(bx)dx[/tex]

[tex]
\int_0^1 dt \ J_0^2(b t) & = & 2\ J_1(b) \sum_{k=0}^{\infty}
\frac{(-1)^k}{2k+1} J_{2k+1}(b).
[/tex]


vietha said:
2. [tex]\int_0^\infty J_0^2(bx)e^{-\frac{x}{c}}dx[/tex]


[tex]
\int_0^\infty dt e^{-pt} J_0^2(bt) & = & \frac{k {\mathbf{K}}(k)}{\pi b}
[/tex]

for

[tex]
Re(p)>0
[/tex]

where
[tex]
k^2 & = & \frac{4 b^2}{p^2 + 4 b^2}
[/tex]
and
[tex]
{\mathbf{K}}(k)}
[/tex]
is the complete elliptic integral of the first kind.



vietha said:
3. [tex]\int_0^\infty J_0^2(bx)e^{-\frac{x^2}{c}}dx[/tex]


[tex]
\int_0^\infty dt e^{-p^2t^2} J_0^2(bt) & = & \frac{\Gamma(\frac{1}{2})}{2p}
\ _3F_3 (\frac{1}{2},1,\frac{1}{2}; 1,1,1 | - \frac{b^2}{p^2} ),
[/tex]
[tex]
for Re(p^2)>0
[/tex]

vietha said:
3. [tex]\int_0^\infty J_0^2(bx)\frac{x}{c}dx[/tex]


Are you sure this converges? Given the asymptotic expansion of [tex]J_0[/tex] I'm skeptical.
 
  • #5
Thanks jasonRF for the results. I have that book too. Could you tell me in which parts and pages you found that?

Originally Posted by vietha View Post

3. [tex]\int_0^\infty J_0^2(bx)\frac{x}{c}dx [/tex]Are you sure this converges? Given the asymptotic expansion of LaTeX Code: J_0 I'm skeptical.
I made a mistake with the last one. It should be:
[tex]\int_0^a J_0^2(bx)\frac{x}{c}dx [/tex]
 
Last edited:
  • #6
Hi there,

I found this integral at Gradshteyn:

[tex]\int_{0}^{1} x\, J_{\nu}(\alpha\,x)J_{\nu}(\beta\,x)\,dx = \frac{\beta J_{\nu-1}(\beta)J_{\nu}(\alpha) - \alpha J_{\nu-1}J_{\nu}(\beta)}{\alpha^2 - \beta^2}.[/tex]

Then, taking the limit [tex]\beta\rightarrow\alpha[/tex] you can find

[tex]\int_{0}^{1}x\,J_{\nu}^2(\alpha\,x) dx = -\frac{1}{2\alpha}\left[J_{\nu-1}(\alpha)J_{\nu}(\alpha) + \alpha J_{\nu-1}^{\prime}(\alpha)J_{\nu}(\alpha) - \alpha J_{\nu-1}(\alpha)J_{\nu}^{\prime}(\alpha)\right].[/tex]

I hope it is useful.
 

1. What is the definition of the integral of square of Bessel function?

The integral of square of Bessel function is defined as the area under the curve of the square of the Bessel function, which is a special function that appears in many areas of mathematics and physics. It is denoted as ∫ |Jν(x)|2 dx, where Jν(x) is the Bessel function of the first kind and order ν.

2. What is the significance of the integral of square of Bessel function?

The integral of square of Bessel function is important in many areas of mathematics and physics, such as signal processing, quantum mechanics, and electromagnetics. It is used to solve various types of differential equations and can also be used to represent physical quantities such as the energy of a vibrating membrane.

3. How is the integral of square of Bessel function calculated?

The integral of square of Bessel function can be calculated using various numerical methods, such as the trapezoidal rule or Simpson's rule. It can also be expressed in terms of other special functions, such as the Gamma function and the modified Bessel function.

4. What are the properties of the integral of square of Bessel function?

The integral of square of Bessel function has several important properties, including orthogonality, recurrence relations, and a closed-form solution for certain values of ν. It also has a connection to the Fourier transform and can be used to calculate the mean square value of a function.

5. Are there any applications of the integral of square of Bessel function?

Yes, the integral of square of Bessel function has many practical applications. For example, it is used in digital signal processing for noise reduction and spectral analysis. It is also used in the design of antennas and waveguides in electromagnetics. In addition, it has applications in quantum mechanics for calculating the probability of finding a particle at a specific location.

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