J(X)

Bessel functions are mathematical special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena with circular symmetry or cylindrical symmetry. They are named after the German astronomer and mathematician Friedrich Bessel, who studied them systematically in 1824. Bessel functions are solutions to a particular type of ordinary differential equation: x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 0 , {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0,} where α {\displaystyle \alpha } is a number that determines the shape of the solution. This number is called the order of the Bessel function and can be any complex number. Although the same equation arises for both α {\displaystyle \alpha } and − α {\displaystyle -\alpha } , mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes. The most important cases are when α {\displaystyle \alpha } is an integer or a half-integer. When α {\displaystyle \alpha } is an integer, the resulting Bessel functions are often called cylinder functions or cylindrical harmonics because they naturally arise when solving problems (like Laplace's equation) in cylindrical coordinates. When α {\displaystyle \alpha } is a half-integer, the solutions are called spherical Bessel functions and are used in spherical systems, such as in solving the Helmholtz equation in spherical coordinates.

The Bullet Journal - 2025-10-28T00:00:00.000000Z

Hardrive Archives - 2025-10-13T00:00:00.000000Z

The Return Of Chedini - 2025-08-25T00:00:00.000000Z

Que lofi ni que la chingada - 2025-06-30T00:00:00.000000Z

Blufaces 2 - 2025-04-20T00:00:00.000000Z

Sounds & Spite - 2024-07-07T00:00:00.000000Z

Sole Beats 2 - 2024-05-17T00:00:00.000000Z

Next to the Pacific - 2020-12-05T00:00:00.000000Z

Rocker Foo - 2020-10-31T00:00:00.000000Z

Light the Candle - 2020-09-30T00:00:00.000000Z

Sounds From The Shed: But You Don't Hear Me Tho - 2020-08-31T00:00:00.000000Z

Las Trailas - 2020-07-31T00:00:00.000000Z

Skat to This - 2020-06-30T00:00:00.000000Z

Short Circuit 2 - 2020-05-30T00:00:00.000000Z

Con La Bendicion De Mi Abuela - 2020-04-30T00:00:00.000000Z

Sole Beats - 2020-03-31T00:00:00.000000Z

Southox - 2020-02-28T00:00:00.000000Z

Ode to the Old School - 2020-01-31T00:00:00.000000Z

Short Circuit - 2019-12-20T00:00:00.000000Z

Kiriyama - 2025-09-16T00:00:00.000000Z

16 Lineas - 2025-08-11T00:00:00.000000Z

In California - 2025-07-15T00:00:00.000000Z

4thaday - 2025-04-09T00:00:00.000000Z

Alladat - 2025-02-10T00:00:00.000000Z

Blake Griffin - 2025-01-14T00:00:00.000000Z