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BesselEquation
The Bessel equation of order
For the Bessel function of the first kind of order
The solution to this differential equation is the Bessel function of the first kind of order 0, denoted by
The Bessel equation can be written as a Sturm-Liouville problem:
with the boundary conditions that the solutions are finite at the origin, and the normalization condition:
The eigenvalues are determined by the zeros of the Bessel function, and the corresponding eigenfunctions are given by:
where
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Eigenvalues: The eigenvalues are given by
$\lambda_n = \alpha_n^2$ where$\alpha_n$ are the positive zeros of the Bessel function$J_0(x)$ . -
Eigenfunctions: The eigenfunctions corresponding to these eigenvalues are
$y_n(x) = J_0(\alpha_n x)$ .
The eigenfunctions are orthogonal with respect to the weight function
This orthogonal property and the known eigenfunctions and eigenvalues provide a complete description of the Hilbert space corresponding to the Bessel function of the first kind of order 0. By employing the series expansion and Sturm-Liouville theory, you can analyze and represent functions in terms of this orthonormal basis, leveraging the well-known properties of Bessel functions.