Bessel function

The Bessel differential equation is a linear ordinary differential equation of second order. It was named after the German mathematician Friedrich Wilhelm Bessel. Its solutions are called Bessel functions or cylinder functions.

• 3.1 Airy integrals
• 3.2 Hypergeometric function
• 3.3 Asymptotic behavior

• 4.1 Further properties

Bessel's differential equation

The Bessel differential equation is an ordinary linear differential equation of second order defined by

Is defined, with a real or complex number. The solutions are called Bessel functions of order.

According to the operator is a Bessel second order differential operator. Defined by

With it you can Bessel's differential equation by short

Express.

Bessel functions

Generally

The solutions of Bessel 's differential equation are called Bessel functions. They play an important role in physics, since the Bessel differential equation is the radial part of the Laplace equation in cylindrical symmetry. On the Bessel functions one meets, among others, in the study of natural vibrations of a circular membrane or organ pipe, the propagation of water waves in round containers, the heat conduction in rods, the analysis of the frequency spectrum of frequency-modulated signals, the field distribution in the cross section of circular waveguides, the stationary states box of potentials, the power distribution in nuclear reactors and the intensity of light diffraction by circular holes. One counts the Bessel functions because of their varied applications in mathematical physics to special functions.

As a second order differential equation Bessel's differential equation must have two linearly independent solutions. Accordingly, there are different variants of the Bessel functions.

Bessel functions of the first kind Jν

The Bessel functions of the first kind are defined as

Where the gamma function. In the origin () these functions for integer are finite.

For non-integer and linear independent solutions. However, for integer relationship

In this case, the second independent solution is found using the Bessel function of the second kind, which is discussed further below.

Integral representations

For integer can represent the Bessel function of the first kind as integral

Hypergeometric function

The Bessel function of the first kind can be expressed by the hypergeometric function:

This expression is related to a function for Bessel Clifford function with the development of the Bessel function.

Bessel functions of the second kind Yν

The Bessel functions of the second kind (also Weber functions or Neumann functions called ) solve Bessel's differential equation. An alternative name is. For non-integer one can define by

For integer you have to make the border crossing

After execution of the border crossing with the rule of L'Hospital we find that these functions have a logarithmic singularity at the origin:

This is Euler's constant and the harmonic series.

For all beside the Bessel function of the first kind Bessel function of the second kind is a second linearly independent solution.

For integer applies as for the Bessel functions of the first kind, the following relationship

Bessel functions of the third kind hn (1), hn (2)

The Bessel functions of the third kind (also known as Hankel functions ) are linear combinations of Bessel functions of the first and second generic

Where i denotes the imaginary unit. These two functions are linearly independent solutions of Bessel 's differential equation.

Other properties

• For the Bessel functions, and recurrence relations apply:

• For all.

• For all.

Asymptotic behavior

We accept the following expressions, that is real and non- negative. For small arguments, the asymptotic expressions are

For large arguments, one finds

These formulas are exact. Compare this with the spherical Bessel functions below.

Modified Bessel functions

The differential equation

Is solved by Bessel functions with purely imaginary argument. One defines for their solution is usually the modified Bessel functions

The function is also known as the MacDonald function. Other than the " normal" Bessel functions, the modified Bessel functions have not oscillating, but an exponential behavior.

Airy integrals

For the functions and can you specify an integral representation

Hypergeometric function

The modified Bessel function of the first kind can be expressed by the hypergeometric function:

Asymptotic behavior

We assume again that is real and non- negative. For small arguments, one finds

For large arguments we obtain

Spherical Bessel functions

The Helmholtz equation in spherical coordinates by separation of variables leads to the radial equation

After the substitution

We obtain the Bessel differential equation

For the solution of the equation is usually the radial spherical Bessel functions, the spherical Neumann functions and the spherical Hankel functions are defined:

Both and and are linearly independent solutions.

There are alternative representations for

The spherical Bessel and Hankel functions are required for example for the treatment of the spherically symmetric potential well in quantum mechanics.

Other properties

• For the spherical Bessel functions, and recurrence relations apply:

• For the Wronskian

Hankel transform

The Hankel transform is an integral transform that is closely related to the Fourier transform. The integral core of the Hankel transform of the Bessel function of the first kind, that is, the integral operator is

A particular feature of the Hankel transform is that in an algebraic expression ( an multiplication) can be converted to its Bessel operator.

History

Bessel functions were discussed in detail by Bessel in 1824, but also appeared previously in special physical problems on for example in Daniel Bernoulli ( vibration heavy chains 1738), Leonard Euler ( membrane vibration 1764), in celestial mechanics at Joseph -Louis Lagrange (1770 ) and Pierre -Simon Laplace and heat conduction at Joseph Fourier (heat propagation in cylinder 1822) and Siméon Denis Poisson (1823 ).