Bessel function

A solution of Bessel’s differential equation, characterized by its order and argument. Bessel functions arise in wave and diffusion problems with cylindrical or spherical symmetry.

Bessel's differential equation: A second-order linear differential equation whose solutions include Bessel functions. Its coefficients determine the characteristic oscillations and singular behavior of these functions.
Bessel function of the first kind: A Bessel-function solution that is finite at the origin for nonnegative integer order. It supplies the regular radial solution in many cylindrical-domain problems.
Helmholtz equation: A partial differential equation describing time-harmonic waves and spatial eigenmodes. Separating it in cylindrical coordinates produces Bessel equations for radial dependence.
Spherical Bessel function: A radial solution of the wave equation in spherical coordinates, expressible through ordinary Bessel functions of half-integer order. It is the spherical counterpart used when symmetry is around a point rather than an axis.
Order (Bessel function): The parameter that distinguishes families of Bessel functions and appears in their differential equation. Changing the order changes the function’s behavior near the origin and its zeros.
Bessel function of the second kind: A linearly independent Bessel-function solution that is singular at the origin for integer order. It completes the solution basis when the domain excludes the axis or origin.
Laplacian in cylindrical coordinates: The differential operator for divergence of gradient expressed in cylindrical coordinates. Its radial component yields Bessel’s equation after angular separation.
Modified Bessel function: A solution of the modified Bessel equation, typically growing or decaying rather than oscillating on positive real arguments. It arises when the radial equation has the opposite sign from the oscillatory Bessel equation.
Argument (Bessel function): The independent variable at which a Bessel function is evaluated. The argument sets the function’s position along its oscillatory profile.
Hankel function: A complex linear combination of Bessel functions of the first and second kinds. Its outgoing and incoming wave forms suit radiation problems in cylindrical geometry.

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