![Vibrating drum Bessel function](https://upload.wikimedia.org/wikipedia/commons/b/bc/Vibrating_drum_Bessel_function.gif)
Bessel functions, first defined by the mathematician
Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...
and then generalized by
Friedrich Bessel
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method ...
, are canonical solutions of Bessel's
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
for an arbitrary
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, the ''order'' of the Bessel function. Although
and
produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly
smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s of
.
The most important cases are when
is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
or
half-integer
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
. Bessel functions for integer
are also known as cylinder functions or the
cylindrical harmonics In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, \nabla^2 V = 0, expressed in cylindrical coordinates, ''ρ'' (radial coordinate), ''φ'' (polar angle), an ...
because they appear in the solution to
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
in
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
.
Spherical Bessel functions with half-integer
are obtained when the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
is solved in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
.
Applications of Bessel functions
The Bessel function is a generalization of the sine function. It can be interpreted as the vibration of a string with variable thickness, variable tension (or both conditions simultaneously); vibrations in a medium with variable properties; vibrations of the disc membrane, etc.
Bessel's equation arises when finding separable solutions to
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
and the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
in cylindrical or
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
. Bessel functions are therefore especially important for many problems of
wave propagation
Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation.
With respect to the direction of the oscillation relative to ...
and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (); in spherical problems, one obtains half-integer orders (). For example:
*
Electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
in a cylindrical
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
* Pressure amplitudes of
inviscid
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
rotational flows
*
Heat conduction
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a te ...
in a cylindrical object
* Modes of vibration of a thin circular or annular
acoustic membrane
An acoustic membrane is a thin layer that vibrates and is used in acoustics to produce or transfer sound, such as a drum, microphone, or loudspeaker.
See also
* Membranophone
* Vibrations of a circular membrane
A two-dimensional elastic ...
(such as a
drumhead
A drumhead or drum skin is a membrane stretched over one or both of the open ends of a drum. The drumhead is struck with sticks, mallets, or hands, so that it vibrates and the sound resonates through the drum.
Additionally outside of percus ...
or other
membranophone
A membranophone is any musical instrument which produces sound primarily by way of a vibrating stretched membrane. It is one of the four main divisions of instruments in the original Hornbostel-Sachs scheme of musical instrument classification. ...
) or thicker plates such as sheet metal (see
Kirchhoff–Love plate theory
The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and w ...
,
Mindlin–Reissner plate theory
The Uflyand-Mindlin theory of vibrating plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1948 by Yakov Solomonovich UflyandUflyand, Ya. ...
)
* Diffusion problems on a lattice
* Solutions to the radial
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
(in spherical and cylindrical coordinates) for a free particle
* Solving for patterns of acoustical radiation
* Frequency-dependent friction in circular pipelines
* Dynamics of floating bodies
*
Angular resolution
Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. ...
* Diffraction from helical objects, including
DNA
*
Probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of product of two normally distributed random variables
* Analyzing of the surface waves generated by microtremors, in
geophysics
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
and
seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
.
Bessel functions also appear in other problems, such as signal processing (e.g., see
FM audio synthesis,
Kaiser window
The Kaiser window, also known as the Kaiser–Bessel window, was developed by James Kaiser at Bell Laboratories. It is a one-parameter family of window functions used in finite impulse response filter design and spectral estimation, spectral analy ...
, or
Bessel filter
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat Group delay and phase delay, group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in ...
).
Definitions
Because this is a second-order linear differential equation, there must be two
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.
Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by and , respectively, rather than and .
Bessel functions of the first kind:
Bessel functions of the first kind, denoted as , are solutions of Bessel's differential equation. For integer or positive , Bessel functions of the first kind are finite at the origin (); while for negative non-integer , Bessel functions of the first kind diverge as approaches zero. It is possible to define the function by its
series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division) ...
around , which can be found by applying the
Frobenius method
In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form
z^2 u'' + p(z)z u'+ q(z) u = 0
with u' \equiv \frac and u'' ...
to Bessel's equation:
[Abramowitz and Stegun]
p. 360, 9.1.10
where is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, a shifted generalization of the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
function to non-integer values. The Bessel function of the first kind is an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
if is an integer, otherwise it is a
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to
(see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large . (The series indicates that is the derivative of , much like is the derivative of ; more generally, the derivative of can be expressed in terms of by the identities
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
.)
For non-integer , the functions and are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order , the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):
This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.
Bessel's integrals
Another definition of the Bessel function, for integer values of , is possible using an integral representation:
which is also called Hansen-Bessel formula.
This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for :
Relation to hypergeometric series
The Bessel functions can be expressed in terms of the
generalized hypergeometric series
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
as
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 funct ...
.
Relation to Laguerre polynomials
In terms of the
Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:
xy'' + (1 - x)y' + ny = 0
which is a second-order linear differential equation. This equation has nonsingular solutions only ...
and arbitrarily chosen parameter , the Bessel function can be expressed as
Bessel functions of the second kind:
The Bessel functions of the second kind, denoted by , occasionally denoted instead by , are solutions of the Bessel differential equation that have a singularity at the origin () and are
multivalued. These are sometimes called Weber functions, as they were introduced by , and also Neumann functions after
Carl Neumann
Carl Gottfried Neumann (also Karl; 7 May 1832 – 27 March 1925) was a German mathematician.
Biography
Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann (1798–1895), who w ...
.
For non-integer , is related to by
In the case of integer order , the function is defined by taking the limit as a non-integer tends to :
If is a nonnegative integer, we have the series
where
is the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strictly ...
, the
logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f''; ...
of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
There is also a corresponding integral formula (for ):
[Watson]
p. 178
In the case where ,
is necessary as the second linearly independent solution of the Bessel's equation when is an integer. But has more meaning than that. It can be considered as a "natural" partner of . See also the subsection on Hankel functions below.
When is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:
Both and are
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s of on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
cut along the negative real axis. When is an integer, the Bessel functions are
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
s of . If is held fixed at a non-zero value, then the Bessel functions are entire functions of .
The Bessel functions of the second kind when is an integer is an example of the second kind of solution in
Fuchs's theorem
In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form
y'' + p(x)y' + q(x)y = g(x)
has a solution expressible by a generalised Frobenius series when p(x), q(x) and g(x) are analytic ...
.
Hankel functions: ,
Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, and , defined as
where is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named after
Hermann Hankel
Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix.
Biography
Hankel was born on 1 ...
.
These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form . For real
where
,
are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
, substituting , for
and
,
for
,
, as explicitly shown in the
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
.
The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the
sign convention
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
for the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
).
Using the previous relationships, they can be expressed as
If is an integer, the limit has to be calculated. The following relationships are valid, whether is an integer or not:
In particular, if with a nonnegative integer, the above relations imply directly that
These are useful in developing the spherical Bessel functions (see below).
The Hankel functions admit the following integral representations for :
where the integration limits indicate integration along a
contour
Contour may refer to:
* Contour (linguistics), a phonetic sound
* Pitch contour
* Contour (camera system), a 3D digital camera system
* Contour, the KDE Plasma 4 interface for tablet devices
* Contour line, a curve along which the function has a ...
that can be chosen as follows: from to 0 along the negative real axis, from 0 to along the imaginary axis, and from to along a contour parallel to the real axis.
Modified Bessel functions: ,
The Bessel functions are valid even for
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
arguments , and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as
when is not an integer; when is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments . The series expansion for is thus similar to that for , but without the alternating factor.
can be expressed in terms of Hankel functions:
Using these two formulae the result to
+
, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following
given that the condition is met. It can also be shown that
only when , , < and but not when .
We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):
and are the two linearly independent solutions to the modified Bessel's equation:
Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, and are
exponentially growing and
decaying functions respectively. Like the ordinary Bessel function , the function goes to zero at for and is finite at for . Analogously, diverges at with the singularity being of logarithmic type for , and otherwise.
Two integral formulas for the modified Bessel functions are (for ):
Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example:
It can be proven by showing equality to the above integral definition for . This is done by integrating a closed curve in the first quadrant of the complex plane.
Modified Bessel functions and can be represented in terms of rapidly convergent integrals
The modified Bessel function
is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.
The modified Bessel function of the second kind has also been called by the following names (now rare):
* Basset function after
Alfred Barnard Basset
* Modified Bessel function of the third kind
* Modified Hankel function
* Macdonald function after
Hector Munro Macdonald
Prof Hector Munro Macdonald FRAS FRSE LLD (19 January 1865 – 16 May 1935) was a Scottish mathematician, born in Edinburgh in 1865. He researched pure mathematics at Cambridge University after graduating from Aberdeen University with an hono ...
Spherical Bessel functions: ,
![Spherical Bessel y Functions (n=0,1,2)](https://upload.wikimedia.org/wikipedia/commons/9/93/Spherical_Bessel_y_Functions_%28n%3D0%2C1%2C2%29.svg)
When solving the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
in spherical coordinates by separation of variables, the radial equation has the form
The two linearly independent solutions to this equation are called the spherical Bessel functions and , and are related to the ordinary Bessel functions and by
is also denoted or ; some authors call these functions the spherical Neumann functions.
From the relations to the ordinary Bessel functions it is directly seen that:
The spherical Bessel functions can also be written as (Rayleigh's formulas)
The zeroth spherical Bessel function is also known as the (unnormalized)
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
. The first few spherical Bessel functions are:
and
Generating function
The spherical Bessel functions have the generating functions
Differential relations
In the following, is any of , , , for
Spherical Hankel functions: ,
There are also spherical analogues of the
Hankel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
:
In fact, there are simple closed-form expressions for the Bessel functions of
half-integer
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
order in terms of the standard
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s, and therefore for the spherical Bessel functions. In particular, for non-negative integers :
and is the complex-conjugate of this (for real ). It follows, for example, that and , and so on.
The spherical Hankel functions appear in problems involving
spherical wave
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seis ...
propagation, for example in the
multipole expansion of the electromagnetic field.
Riccati–Bessel functions: , , ,
Riccati
Jacopo Francesco Riccati (28 May 1676 – 15 April 1754) was a Venetian mathematician and jurist from Venice. He is best known for having studied the equation which bears his name.
Education
Riccati was educated first at the Jesuit school for th ...
–Bessel functions only slightly differ from spherical Bessel functions:
They satisfy the differential equation
For example, this kind of differential equation appears in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
while solving the radial component of the
Schrödinger's equation with hypothetical cylindrical infinite potential barrier. 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
The Mie solution to Maxwell's equations (also known as the Lorenz–Mie solution, the Lorenz–Mie–Debye solution or Mie scattering) describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the f ...
after the first published solution by Mie (1908). See e.g., Du (2004) for recent developments and references.
Following
Debye
The debye (symbol: D) (; ) is a CGS unit (a non- SI metric unit) of electric dipole momentTwo equal and opposite charges separated by some distance constitute an electric dipole. This dipole possesses an electric dipole moment whose value is give ...
(1909), the notation , is sometimes used instead of , .
Asymptotic forms
The Bessel functions have the following
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
forms. For small arguments