# Bessel function

modified Bessel functionspherical Bessel functionBesselmodified Bessel function of the second kindmodified Bessel function of the first kindBessel functionsspherical Bessel functionsspherical Hankel functionsBessel function of the first kindmodified Bessel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutionswikipedia
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### Friedrich Bessel

BesselFriedrich Wilhelm BesselBessel, Friedrich
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions
A special type of mathematical functions were named Bessel functions after Bessel's death, though they had originally been discovered by Daniel Bernoulli and then generalised by Bessel.

### Cylindrical harmonics

Bessel functions for integer are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates.
The ρ-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).

### Bessel filter

BesselBessel electronic filterstime-delay networks
Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).
The filters are also called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949.

### Generalized hypergeometric function

generalized hypergeometric series 7 ''F'' 6 hypergeometric seriesconfluent hypergeometric limit function
The Bessel functions can be expressed in terms of the generalized hypergeometric series as
Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.

### Angular resolution

spatial resolutionresolutionresolved
Angular resolution
This number is more precisely 1.21966989..., the first zero of the order-one Bessel function of the first kind J_{1}(x) divided by π.

### Helmholtz equation

inhomogeneous Helmholtz equationHelmholtzHelmholtz equations
Spherical Bessel functions with half-integer are obtained when the Helmholtz equation is solved in spherical coordinates.
where the Bessel function J n satisfies Bessel's equation

### Kaiser window

Kaiser-Bessel derived
Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).
is the zeroth-order modified Bessel function of the first kind,

### Bessel–Clifford function

This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.
In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions.

### Asymptotic analysis

asymptoticasymptoticallyasymptotics
The Bessel functions have the following asymptotic forms.
* Hankel functions

### Frequency modulation synthesis

FM synthesisFMFM synthesizer
Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).
where are angular frequencies of carrier and modulator, is frequency modulation index, and amplitudes is n\,-th Bessel function of first kind, respectively.

### Entire function

The Bessel function of the first kind is an entire function if is an integer, otherwise it is a multivalued function with singularity at zero.
the Bessel function J 0 (z)

### Hankel transform

Fourier–Bessel or Hankel transformHankel
is the rectangle function) then the Hankel transform of it (of any given order
In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind

### Alfred Barnard Basset

Basset, Alfred Barnard
Basset function after Alfred Barnard Basset
He also worked on Bessel functions: the term Basset function was at one time used for modified Bessel functions of the second kind but is now obsolete.

### Neumann polynomial

Neumann's polynomial
where is Neumann's polynomial.
In mathematics, a Neumann polynomial, introduced by Carl Neumann for the special case \alpha=0, is a polynomial in 1/z used to expand functions in term of Bessel functions.

### Hermann Hankel

HankelHankel, Hermann
They are named after Hermann Hankel.
Hankel functions in the theory of Bessel functions

### Euler–Mascheroni constant

Euler's constantEuler gamma constant
where is the Euler–Mascheroni constant (0.5772...).
Solution of the second kind to Bessel's equation

### Plane wave expansion

convertsum of cylindrical waves
which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal.
are spherical Bessel functions,

### Electromagnetic wave equation

The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.
The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions.

### Mie scattering

Mieelectromagnetic plane wave scatteringLorenz-Mie
This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908).
expressed in terms of Riccati-Bessel functions S_L, C_L that can be expressed in terms of spherical Bessel functions j_L, y_L or Bessel functions of non-integer order and follow recurrent relationships

### Fourier–Bessel series

. This orthogonality relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function is expanded in the basis of the functions
In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions.

### Wave equation

spherical wavewavelinear wave equation
The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.
where and are the spherical Hankel functions.

### Frequency modulation

FMfrequency modulatedfrequency-modulated
which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal.
The harmonic distribution of a sine wave carrier modulated by such a sinusoidal signal can be represented with Bessel functions; this provides the basis for a mathematical understanding of frequency modulation in the frequency domain.

### Sinc function

sinccardinal sine functionnormalized sinc function
is also known as the (unnormalized) sinc function.
* The unnormalized sinc is the zeroth-order spherical Bessel function of the first kind,

### Hector Munro Macdonald

Macdonald, Hector Munro
Macdonald function after Hector Munro Macdonald
Macdonald worked on electric waves and solved difficult problems regarding diffraction of these waves by summing series of Bessel functions.

### Abel's identity

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:
It is especially useful for equations such as Bessel's equation where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult to compute directly.