#Fm expressions series#

Frequency-dependent friction in circular pipelines.Solving for patterns of acoustical radiation.Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle.Modes of vibration of a thin circular or annular acoustic membrane (such as a drum or other membranophone) or thicker plates such as sheet metal (see Kirchhoff–Love plate theory, Mindlin–Reissner plate theory).Heat conduction in a cylindrical object.Pressure amplitudes of inviscid rotational flows.Electromagnetic waves in a cylindrical waveguide.In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order ( α = n) in spherical problems, one obtains half-integer orders ( α = n + 1 / 2). Bessel functions are therefore especially important for many problems of wave propagation and static potentials. It can be interpreted as the vibration of a string with variable thickness, variable tension (or both conditions simultaneously) vibrations in a medium with variable properties vibrations of the disc membrane, etc.īessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. The Bessel function is a generalization of the sine function. 2.7 Riccati–Bessel functions: S n, C n, ξ n, ζ nĪpplications of Bessel functions.2.6 Spherical Hankel functions: h (1) n, h (2) n.2.5 Spherical Bessel functions: j n, y n.2.4 Modified Bessel functions: I α, K α.2.2 Bessel functions of the second kind: Y α.

#Fm expressions series#

2.1.2 Relation to hypergeometric series.2.1 Bessel functions of the first kind: J α.