In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation: for an arbitrary real number α (the order). The most common and important special case is where α is an integer, n. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel func ...

Including:

• Bessel function - Applications
• Bessel function - Definitions
• Bessel function - Bessel functions of the first kind
• Bessel function - Bessel functions of the second kind
• Bessel function - Hankel functions
• Bessel function - Modified Bessel functions
• Bessel function - Spherical Bessel functions
• Bessel function - Riccati-Bessel functions
• Bessel function - Asymptotic forms
• Bessel function - Properties

Read more here: » Bessel function: Encyclopedia - Bessel function

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = n; for spherical problems, one obtains half integer orders α = n+½.) For example: electromagnetic waves in a cylindrical waveguide heat conduction in a cylindrical object. modes of vibration ...

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Bessel function, Bessel function - Applications, Bessel function - Definitions, Bessel function - Bessel functions of the first kind, Bessel function - Bessel functions of the second kind, Bessel function - Hankel functions, Bessel function - Modified Bessel functions, Bessel function - Spherical Bessel functions, Bessel function - Riccati-Bessel functions, Bessel function - Asymptotic forms, Bessel function - Properties

Read more here: » Bessel function: Encyclopedia II - Bessel function - Applications

Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below. Bessel function - Bessel functions of the first kind. Bessel functions of the first kind, denoted with Jα(x), are solutions of Bessel's differential equation which are finite at x = 0 for α an integer or α non-negative. The s ...

See also:

Bessel function, Bessel function - Applications, Bessel function - Definitions, Bessel function - Bessel functions of the first kind, Bessel function - Bessel functions of the second kind, Bessel function - Hankel functions, Bessel function - Modified Bessel functions, Bessel function - Spherical Bessel functions, Bessel function - Riccati-Bessel functions, Bessel function - Asymptotic forms, Bessel function - Properties

Read more here: » Bessel function: Encyclopedia II - Bessel function - Definitions

The Bessel functions have the following asymptotic forms. For small arguments 0 < x << 1, one obtains: where α is non-negative, γ is the Euler-Mascheroni constant (0.5772...), and Γ denotes the gamma function. For large arguments x >> 1, they become: Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations. For example, for large x >> 1, the modified Bessel functio ...

See also:

Bessel function, Bessel function - Applications, Bessel function - Definitions, Bessel function - Bessel functions of the first kind, Bessel function - Bessel functions of the second kind, Bessel function - Hankel functions, Bessel function - Modified Bessel functions, Bessel function - Spherical Bessel functions, Bessel function - Riccati-Bessel functions, Bessel function - Asymptotic forms, Bessel function - Properties

Read more here: » Bessel function: Encyclopedia II - Bessel function - Asymptotic forms