SUMMARY
The discussion centers on proving the Fourier Bessel series expansion represented by the equation \(\frac{1}{\left(\rho^2+z^2\right)^{1/2}}=\int^{\infty}_{0} e^{-k\left|z\right|}J_{0}(k\rho)dk\). Participants emphasize the importance of using the integral representation of the zero-order Bessel function \(J_{0}(k\rho)\) and suggest switching the order of integration to facilitate the proof. The connection between the Hankel transform and the 2D Fourier transform of cylindrical symmetric functions is also highlighted as a crucial step in the solution.
PREREQUISITES
- Understanding of Fourier Bessel series and their applications
- Familiarity with Bessel functions, specifically the zero-order Bessel function \(J_{0}(k\rho)\)
- Knowledge of integral transforms, particularly the Hankel transform
- Basic principles of 2D Fourier transforms and cylindrical symmetry
NEXT STEPS
- Study the properties and applications of Bessel functions in mathematical physics
- Learn about the Hankel transform and its relation to Fourier transforms
- Explore the derivation and applications of the 2D Fourier transform
- Investigate limiting procedures in mathematical proofs and their significance
USEFUL FOR
Mathematicians, physicists, and engineering students focusing on applied mathematics, particularly those working with Fourier analysis and Bessel functions in cylindrical coordinates.