Integration of the product of sine and the first Bessel function

In summary, the task is to prove that the integral of sine times the first Bessel function over an infinite interval equals the square root of (a^2 - p^2) if p is less than a, and equals 0 if p is greater than a. The person has attempted to use the inverse order of integration and the delta Dirac function, as well as the series form of both sine and the Bessel function, but so far has not been successful. They are requesting advice on how to solve this problem.
  • #1
Ardat
2
0

Homework Statement



I'm supposed to prove that:

[itex]\int_0^∞[STRIKE][/STRIKE][/itex]sin(ka)J0(kp)dk = (a2 - p2)1/2 if p < a
and = 0 if p > a

J0 being the first Bessel function.

Homework Equations





The Attempt at a Solution



I've tried to inverse the order of integration and then make the integral form of the delta Dirac function appear, but I'm not sure how to do it, and so far my attemps have failed.

I also tried to put both sine and the bessel function as their series form, then transfom the infinite series into a limit of a finite series so I can interchange the sum and the integral, but it doesn't really leads me anywhere.

If anyone could give me some advise on how to resolve this, I would be grateful.
 
Physics news on Phys.org
  • #2
Nobody can help me?
 

1. What is the product of sine and the first Bessel function?

The product of sine and the first Bessel function, denoted as sin(x)J1(x), is a mathematical function that represents the multiplication of the sine function and the first order Bessel function. It is commonly used in the study of wave phenomena and oscillatory systems.

2. What is the significance of integrating the product of sine and the first Bessel function?

The integration of sin(x)J1(x) has various applications in physics and engineering, particularly in the analysis of vibrating systems and wave propagation. It also plays a crucial role in signal processing and image reconstruction techniques.

3. How is the integration of sin(x)J1(x) performed?

The integration of sin(x)J1(x) can be done analytically using techniques such as integration by parts or by using special functions. It can also be evaluated numerically using numerical integration methods, such as the trapezoidal rule or Simpson's rule.

4. What are the properties of the integrated product of sine and the first Bessel function?

The integrated product of sin(x)J1(x) has several properties, including being an oscillatory function with an infinite number of zeros, and its value at any point is dependent on the value of the sine function and the first Bessel function at that point. It also has a specific relation to the zeros of the first Bessel function.

5. What are some real-world applications of sin(x)J1(x)?

The product of sine and the first Bessel function has various practical applications, including in the analysis of wave phenomena such as sound and light waves, in the design of acoustic and electromagnetic devices, and in the solution of differential equations in physics and engineering.

Similar threads

Question about the Frobenius method and Bessel functions
    • Calculus and Beyond Homework Help
Replies
4
Views
2K
Symbolic integration of a Bessel function with a complex argument
    • MATLAB, Maple, Mathematica, LaTeX
Replies
5
Views
1K
I What is the indefinite integral of Bessel function of 1 order (first k
    • Calculus
Replies
5
Views
1K
An integral of Bessel functions
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Integral simplification using Bessel functions
    • Calculus and Beyond Homework Help
Replies
8
Views
2K
I Integration of Bessel function products (J_1(x)^2/xdx)
    • Quantum Physics
Replies
1
Views
768
Differential Equation with Bessel Function
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Integrating \int xJ_0(ax)J_0(bx)dx w/ Bessel Functions
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
A Integration of Bessel's functions
    • Calculus
Replies
4
Views
1K
Simple integration of bessel functions
    • Calculus and Beyond Homework Help
Replies
2
Views
4K

Hot Threads

Recent Insights

Back
Top