Bessel function is a mathematical function that is widely used in physics and engineering. It is named after the German mathematician Friedrich Bessel, who first introduced it in 1817. The Bessel function is a solution to a second-order linear differential equation that arises in many physical applications, such as the wave equation, heat conduction equation, and diffusion equation.
Definitions
The Bessel function is a family of functions that are solutions to the Bessel differential equation. This equation arises in many physical problems involving cylindrical symmetry, such as the propagation of sound waves in a pipe or the diffusion of heat in a cylinder. The Bessel function is defined as:
J_n(x) = (1/π) ∫_0^π [cos(x sinθ – nθ)] dθ.
where J_n(x) is the Bessel function of order n and argument x, and θ is the angle between x and the positive x-axis.
Origin
The Bessel function was first introduced by Friedrich Bessel in 1817 while studying the problem of the motion of a planet around the Sun. He found that the solution to the differential equation that described the motion of the planet was expressed in terms of a series of functions that are now known as Bessel functions.
Meaning in different dictionaries
According to the Oxford English Dictionary, the Bessel function is “a solution of a differential equation of the second order that arises in the theory of wave motion and other physical problems.” The Merriam-Webster Dictionary defines it as “a mathematical function that is a solution of a differential equation and that is used especially in physics to describe wave motion and other phenomena.”
Associations
The Bessel function is associated with many physical phenomena, such as the propagation of sound waves in a cylindrical pipe, the diffraction of light by a circular aperture, and the flow of heat in a cylinder. It is also used in the analysis of electromagnetic fields, quantum mechanics, and signal processing.
Synonyms
The Bessel function is also known as the cylindrical function, the Bessel series, and the Bessel polynomial.
Antonyms
There are no antonyms of Bessel function as it is a mathematical function with no opposite or negative counterpart.
The same root words
The Bessel function is named after the German mathematician Friedrich Bessel, who first introduced it in 1817. The word “Bessel” is derived from the Old High German word “bizzil,” which means “pointed tool” or “awl.”
Example Sentences
- The Bessel function is widely used in physics and engineering to describe wave motion and other physical phenomena.
- The Bessel function of order zero is also known as the zeroth-order Bessel function or the cylindrical function of the first kind.
- The Bessel function has many applications in signal processing, such as the analysis of digital filters and the design of wavelets.
- The Bessel function is a special case of the hypergeometric function, which is a generalization of many other mathematical functions.
- The Bessel function is sometimes called the Bessel series because it can be expressed as an infinite sum of terms.