Bessel differential equation is widely used in many fields of physics, such as wave propagation in cylindrical coordinates, solution of Schrodinger equation, membrane vibration, heat conduction and so on. The following summarizes some properties of Bessel function without proof, and the related proof is complicated. Please refer to relevant monographs, such as Methods of Mathematical Physics-Wu et al. Mathematical physics method-Gu Qiao; Mathematic methods of physicists-—ArfkenandWeber.

Bessel function is the solution of Bessel equation, which is combined with other functions to form cylindrical harmonic function. Besides elementary function, Bessel function is the most commonly used function in physics and engineering. They were named after F.W. Bessel, a German astronomer in the19th century, who first described them in 1824.

The distribution of drum head amplitude along the radial direction is Bessel function (considering the sign). In fact, these vibrations are the superposition of Bessel functions of various orders.

basic concept

Is the floorboard of a special function in mathematics. The solutions of these equations cannot be expressed systematically by elementary functions. The specific form of Bessel function changes with the change of any real number in the above formula (correspondingly called the order of its corresponding Bessel function).

The most common case in practical application is integer, and the corresponding solution is called n-order Bessel function. Although its own symbol does not change the form of the equation in the above differential equation, it is still used to defining two different Bessel functions in practical application (which can bring benefits, such as eliminating the unsmooth points of the functions).

Functional image

Features:

1, the first Bessel function of odd order is odd function; Even order is an even function. (Odd is odd, even is even)

2. The maximum value of Bessel function is less than or equal to 1. When x tends to infinity, its value tends to 0, and there are infinitely many zeros.