Bessel Functions Bessel functions, are canonical solutions y(x) of Bessel's differential equation: α (the order of the Bessel function) Bessel functions.

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Presentation transcript:

Bessel Functions Bessel functions, are canonical solutions y(x) of Bessel's differential equation: α (the order of the Bessel function) Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates.

Applications of Bessel functions Electromagnetic waves in a cylindrical waveguide 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

Bessel functions of the first kind : Jα

Bessel functions of the second kind : Yα

Circular membrane

Modified Bessel Equation

Modified Bessel functions : Iα, Kα

Modified Bessel functions : Iα, Kα