|
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions ''y''(''x'') of Bessel's differential equation : for an arbitrary complex number α (the order of the Bessel function). Although α and −α produce the same differential equation for real α, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α. The most important cases are for α an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates. ==Applications of Bessel functions== Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. 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). For example: * Electromagnetic waves in a cylindrical waveguide * Pressure amplitudes of inviscid rotational flows * Heat conduction in a cylindrical object * Modes of vibration of a thin circular (or annular) artificial membrane (such as a drum or other membranophone) * Diffusion problems on a lattice * Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle * Solving for patterns of acoustical radiation * Frequency-dependent friction in circular pipelines * Dynamics of floating bodies * Angular resolution Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bessel function」の詳細全文を読む スポンサード リンク
|