On Bessel function's orthogonality

In summary, using the orthogonality relation of Bessel functions, we can determine that the first integral, \displaystyle\int_0^1J_1(x)xJ_2(x)dx, is not necessarily zero. However, for the second integral, \displaystyle\int_0^1J_1(k_1x)J_1(k_2x)dx, where k_1 and k_2 are two distinct zeros of Bessel function of order 1, we can argue that it must be non-zero since \displaystyle\int_0^1J_1(k_1x)xJ_1(k_2x)dx is zero. This conclusion is based on the
  • #1
samuelandjw
22
0
Use the orthogonality relation of Bessel function to argue whether the following two integrals are zero or not:
[itex]\displaystyle\int_0^1J_1(x)xJ_2(x)dx[/itex]
[itex]\displaystyle\int_0^1J_1(k_1x)J_1(k_2x)dx[/itex], where [tex]k_1,k_2[/tex] are two distinct zeros of Bessel function of order 1.

The textbook we are using is Boas's Mathematical Methods in the Physical Sciences, so the formulas that are available to us are a set of recursion relations of Bessel function, and the orthogonality relations between Bessel function of the same order: http://upload.wikimedia.org/math/c/d/b/cdb1e8ba98f7855eba9777024cce03fd.png.

For the first integral, the two Bessel functions are of different order, and there is no zeros in the arguments of the two functions, so I have no idea how to link the first integral to the orthogonality relation of Bessel functions.

For the second integral, my argument is that since [itex]\displaystyle\int_0^1J_1(k_1x)xJ_1(k_2x)dx[/itex] is zero, the 2nd integral (note that there is no x between the two Bessel function) cannot be zero. I think this argument is quite weak. Would anyone give me a better argument?

Thanks.
 
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  • #2
The first one integral(positive function)=positive number
 
Last edited:
  • #3
lurflurf said:
The first one integral(positive function)=positive number

Thanks for your reply. We can surely say that [tex]J_1(x),J_2(x)[/tex] are positive functions between x=0 and x=1 because the first nontrivial zeros are larger than 1. One has to somehow use the information of the location of zeros to reach this conclusion. Suppose we don't have this information, is it still possible to argue that the integral is non-zero?
 

What is the definition of Bessel functions?

Bessel functions are a family of special functions that arise in many mathematical and scientific applications, particularly in problems involving wave phenomena and circular or cylindrical symmetry.

What is the orthogonality property of Bessel functions?

The orthogonality property of Bessel functions states that the integral of the product of two Bessel functions with different indices over a specified interval is equal to zero, unless the indices are equal. This property is useful in solving differential equations and in the Fourier analysis of periodic functions.

What is the significance of the orthogonality property in applications?

The orthogonality property of Bessel functions is important in various areas of science and engineering, including acoustics, electromagnetics, and heat transfer. It allows us to express functions as infinite series of Bessel functions, making it easier to analyze and solve problems involving wave phenomena and cylindrical symmetry.

What are the applications of Bessel functions in physics?

Bessel functions have numerous applications in physics, including in the study of heat conduction, fluid dynamics, and quantum mechanics. They are also used in the analysis of vibrating strings and circular membranes, as well as in the diffraction of light and sound waves.

How are Bessel functions related to other special functions?

Bessel functions are closely related to other special functions, such as the Legendre polynomials and the hypergeometric functions. They can also be expressed in terms of other special functions, such as the exponential and trigonometric functions, through various transformation formulas.

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