A Fourier Bessel series expansion is a mathematical method used to represent a function as an infinite sum of sine and cosine functions multiplied by Bessel functions. It is commonly used to approximate functions that are defined on a disk or a sphere.
Proving the validity of Fourier Bessel series expansion using limiting procedures ensures that the series accurately represents the original function, and that the coefficients of the series converge to the true values. This is important for the reliability and accuracy of any calculations or applications using the series.
The key steps in proving Fourier Bessel series expansion with limiting procedures involve first defining the Bessel functions and their properties, then using these properties to derive the Fourier Bessel series expansion formula. The formula is then tested by taking limits and comparing the resulting series to the original function.
Fourier Bessel series expansion is limited to functions that are defined on a disk or a sphere. It may also be difficult to calculate the coefficients of the series for functions with complicated or discontinuous behavior.
Fourier Bessel series expansion is commonly used in physics and engineering to model and solve problems involving circular or spherical symmetry, such as heat transfer, fluid dynamics, and electromagnetism. It is also used in image and signal processing to compress and analyze data.