What is the Bessel Function?
Bessel functions (named after the astronomer F.W. Bessel) are solutions to differential equations:
Where:
- n is a non-negative real number.
Function values don’t usually have to be calculated by hand; They can be found in many tables (like these Bessel tables). The solutions are called Bessel functions of order n or—less commonly—cylindrical functions of order n. They are one of the most widely used functions in applied mathematics and are popular in problems that involve circular or cylindrical symmetry, so are sometimes called cylinder functions. They are also important in the study of wave propagation [1]. Bessel functions appear in probability theory and statistics in various contexts. Leveraging their properties proves to be a valuable tool, for instance, in calculating important multivariate statistics [2], or understanding diffusion processes [3]. Plus, applying probability theory techniques with distributions containing Bessel functions leads to the derivation of numerous properties of these functions [4]. Notable examples of this approach can be found in [5], [6], and [7]. Scientific fields that frequently use these functions include acoustic theory, electric field theory, hydrodynamics, nuclear physics, and radio physics.
Solutions to Bessel’s Equation
Bessel’s equation is a second-order differential equation with two linearly independent solutions. A second-order differential equation has two independent variables, where the highest order derivative is the second derivative. In the context of Bessel’s equation, the independent variable is x, and the second derivative is d2y/dx2.
Bessel Function of the first kind
Bessel functions of the first kind (sometimes called ordinary Bessel functions), are denoted by Jn(x), where n is the order. 
Bessel Function of the second kind
The second solution (Yv or Nv) is called a Bessel Function of the second kind and is denoted by nn(x). It can also be expressed as a Neumann function: 
The Bessel function of the third kind, also called a Hankel function or Weber function is a complex-valued solution to Bessel’s differential equation. Essentially, the Bessel function of the third kind is a combination of Bessel functions of the first and second kind.
Uses
A large number of fields use Bessel functions, including:
- Acoustic theory,
- Electric field theory,
- Hydrodynamics,
- Nuclear Physics,
- Radio Physics.
History
Although the functions are named after Bessel [8] they appear in much earlier work, including:
- Euler’s 1760’s work on vibrations of a stretched membrane [9],
- Fourier’s 1822 theory of heat flow in spherical bodies [10].
Bernoulli (1703) solved a differential equation by an infinite series, which is largely regarded as the first time the functions appeared in print. It was Bessel, however, who studied the functions in detail while investigating the elliptic motion of planets.
Hankel Function
The Hankel function is a complex-valued solution to Bessel’s differential equation. These functions are very useful for problems involving spherical and cylindrical wave propagation. It is also called a Bessel function of the third kind, or a Weber Function. In essence, a Hankel Function is a combination of Bessel functions of the first kind and second kind. So you can think of it as a “type” of Bessel function. In many mathematical programs, the Hankel is defined in terms of the Bessel. For example, in MATLAB, the Hankel function syntax is:
- First Kind: H = besselh(nu,Z),
- First or Second Kind: H = besselh(nu,K,Z).
Where:
- k = each element (1 or 2) of the complex array z,
- nu = the order of the Hankel function. This must be the same size as Z, or one can be scalar. For example, besselh(4,Z).
As far as evaluating the function, you’re probably going to want to use software, because it is notoriously difficult to evaluate numerically by hand, especially for large order and large argument [11].
Hankel Function of the First Kind
The Hankel function of the first kind is defined as: 
Where:
- Jn(z) = Bessel function of the first kind,
- Yn(z) = Bessel function of the second kind.
Contour Integral Definition
The Hankel function can also be represented by the following contour integral: 
Real Life Applications
Bessel function theory (and the associated Hankel functions) have many real life applications, including:
- Acoustics,
- Atomic and nuclear physics,
- Hydrodynamics,
- Radio physics.
These functions are especially useful for problems related to spherical and cylindrical wave propagation. When it comes to evaluating the function, using software is preferable because it is notoriously challenging to evaluate numerically by hand, particularly for large orders and arguments [11].
References
[1] NIST. Bessel Functions applications. Retrieved August 14, 2023 from: https://dlmf.nist.gov/10.73
[2] Miller, K.S. – Multidimensional Gaussian distributions, John Wiley, New York, London etc. (1964).
[3] Ito, K. and H.P. McKean Jr. – Diffusion processes and their sample paths, Springer, Berlin Heidelberg-New York (1974)
[4] Stadje, J. Mathematics. Probabilistic proofs of some formulas for Bessel functions. Proceedings A 86 (3), September 26, 1983. Elsevier.
[5] Feller, W. – An introduction to probability theory and its applications, Volume II, John Wiley, New York, London etc. (1971). (pp 58-61,479-482)
[6] Feller, W. – Infinitely divisible distributions and Bessel functions associated with random walks, J. Sot. Indust. Appl. Math. 14, 864-875 (1966).
[7] Kent, J. – Some probabilistic properties of Bessel functions, Ann. Prob. 6, 760-770 (1978)
[8] Bessel, F. (1825). Uber die Berechnung der geo-graphischen Längen und Breiten aus geodätischen Vermessungen (The calculation of longitude and latitude from geodesic measurements), Astronomische Nachrichten, 4, 241-254.
[9] Euler, L. (1766). De motu vibratorio tympanorum, Novi Commentarii academiae scientiarum Petropolitanae. 10, 1766, pp. 243-260.
[10] Fourier, M. 1822. Theorie Analytique De La Chaleur.
[11]Jentschura, U. Numerical calculation of Bessel, Hankel and Airy functions. Retrieved November 30, 2019 from: http://arxiv-export-lb.library.cornell.edu/pdf/1112.0072