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Bessel functions, first defined by the mathematician Daniel Bernoulli 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 ...
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 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 fo ...
\alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha 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 functions of \alpha. The most important cases are when \alpha 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 languag ...
or half-integer. Bessel functions for integer \alpha 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
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 d ...
. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation 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'' meas ...
.


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 and the Helmholtz equation 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'' meas ...
. 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 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 rotational flows * Heat conduction 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 A membranophone is any musical instrument which produces sound ...
(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 percu ...
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 ...
,
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, Y ...
) * 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 th ...
(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 * 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) ca ...
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'' so ...
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 analysis. The Kaiser wi ...
, or
Bessel filter In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in the passband. Bessel filters ...
).


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 divisi ...
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
J_\alpha(x) = \sum_^\infty \frac ^, 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 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 x^ (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.) 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): J_(x) = (-1)^n J_n(x). 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: J_n(x) = \frac \int_0^\pi \cos (n \tau - x \sin \tau) \,d\tau = \frac \int_^\pi e^ \,d\tau, 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 : J_\alpha(x) = \frac \int_0^\pi \cos(\alpha\tau - x \sin\tau)\,d\tau - \frac \int_0^\infty e^ \, dt.


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, wh ...
as J_\alpha(x) = \frac \;_0F_1 \left(\alpha+1; -\frac\right). This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.


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 on ...
and arbitrarily chosen parameter , the Bessel function can be expressed as \frac = \frac \sum_^\infty \frac \frac.


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. For non-integer , is related to by Y_\alpha(x) = \frac. In the case of integer order , the function is defined by taking the limit as a non-integer tends to : Y_n(x) = \lim_ Y_\alpha(x). If is a nonnegative integer, we have the series Y_n(z) =-\frac\sum_^ \frac\left(\frac\right)^k +\frac J_n(z) \ln \frac -\frac\sum_^\infty (\psi(k+1)+\psi(n+k+1)) \frac where \psi(z) 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 strict ...
, 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
Y_n(x) = \frac \int_0^\pi \sin(x \sin\theta - n\theta) \, d\theta -\frac \int_0^\infty \left(e^ + (-1)^n e^ \right) e^ \, dt. In the case where , Y_\left(x\right)=\frac\int_^\cos\left(x\cos\theta\right)\left(e+\ln\left(2x\sin^2\theta\right)\right)\, d\theta. 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: Y_(x) = (-1)^n Y_n(x). Both and are holomorphic functions of on the complex plane 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 analyt ...
.


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 \begin H_\alpha^(x) &= J_\alpha(x) + iY_\alpha(x), \\ H_\alpha^(x) &= J_\alpha(x) - iY_\alpha(x), \end 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 ...
. 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 x>0 where J_\alpha(x), Y_\alpha(x) 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 fo ...
, substituting , for e^ and J_\alpha(x), Y_\alpha(x) for \cos(x), \sin(x), 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 \begin H_\alpha^(x) &= \frac, \\ H_\alpha^(x) &= \frac. \end If is an integer, the limit has to be calculated. The following relationships are valid, whether is an integer or not: \begin H_^(x) &= e^ H_\alpha^ (x), \\ H_^(x) &= e^ H_\alpha^ (x). \end In particular, if with a nonnegative integer, the above relations imply directly that \begin J_(x) &= (-1)^ Y_(x), \\ Y_(x) &= (-1)^m J_(x). \end These are useful in developing the spherical Bessel functions (see below). The Hankel functions admit the following integral representations for : \begin H_\alpha^(x) &= \frac\int_^ e^ \, dt, \\ H_\alpha^(x) &= -\frac\int_^ e^ \, dt, \end 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 ha ...
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 \begin I_\alpha(x) &= i^ J_\alpha(ix) = \sum_^\infty \frac\left(\frac\right)^, \\ K_\alpha(x) &= \frac \frac, \end 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. K_ can be expressed in terms of Hankel functions: K_(x) = \begin \frac i^ H_\alpha^(ix) & -\pi < \arg x \leq \frac \\ \frac (-i)^ H_\alpha^(-ix) & -\frac < \arg x \leq \pi \end Using these two formulae the result to J_^2(z)+Y_^2(z), commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following J_^2(x)+Y_^2(x)=\frac\int_^\cosh(2\alpha t)K_0(2x\sinh t)\, dt, given that the condition is met. It can also be shown that J_^2(x)+Y_^2(x)=\frac\int_^K_(2x\sinh t)\, dt, 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 ): \begin J_\alpha(iz) &= e^ I_\alpha(z), \\ Y_\alpha(iz) &= e^I_\alpha(z) - \frace^K_\alpha(z). \end and are the two linearly independent solutions to the modified Bessel's equation: x^2 \frac + x \frac - \left(x^2 + \alpha^2 \right)y = 0. 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 ): \begin I_\alpha(x) &= \frac\int_0^\pi e^ \cos \alpha\theta \,d\theta - \frac\int_0^\infty e^ \,dt, \\ K_\alpha(x) &= \int_0^\infty e^ \cosh \alpha t \,dt. \end Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example: 2\,K_0(\omega) = \int_^\infty \frac \,dt. 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 \begin K_(\xi) &= \sqrt \int_0^\infty \exp \left(- \xi \left(1+\frac\right) \sqrt \right) \,dx, \\ K_(\xi) &= \frac \int_0^\infty \frac \exp \left(- \xi \left(1+\frac\right) \sqrt\right) \,dx. \end The modified Bessel function K_(\xi)=\xi^\exp(-\xi) 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 ...


Spherical Bessel functions: ,

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form x^2 \frac + 2x \frac + \left(x^2 - n(n + 1)\right) y = 0. 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 \begin j_n(x) &= \sqrt J_(x), \\ y_n(x) &= \sqrt Y_(x) = (-1)^ \sqrt J_(x). \end 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: \begin j_n(x) &= (-1)^ y_ (x) \\ y_n(x) &= (-1)^ j_(x) \end The spherical Bessel functions can also be written as (Rayleigh's formulas) \begin j_n(x) &= (-x)^n \left(\frac\frac\right)^n \frac, \\ y_n(x) &= -(-x)^n \left(\frac\frac\right)^n \frac. \end The zeroth spherical Bessel function is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: \begin j_0(x) &= \frac. \\ j_1(x) &= \frac - \frac, \\ j_2(x) &= \left(\frac - 1\right) \frac - \frac, \\ j_3(x) &= \left(\frac - \frac\right) \frac - \left(\frac - 1\right) \frac \end and \begin y_0(x) &= -j_(x) = -\frac, \\ y_1(x) &= j_(x) = -\frac - \frac, \\ y_2(x) &= -j_(x) = \left(-\frac + 1\right) \frac - \frac, \\ y_3(x) &= j_(x) = \left(-\frac + \frac\right) \frac - \left(\frac - 1\right) \frac. \end


Generating function

The spherical Bessel functions have the generating functions \begin \frac \cos \left(\sqrt\right) &= \sum_^\infty \frac j_(z), \\ \frac \sin \left(\sqrt\right) &= \sum_^\infty \frac y_(z). \end


Differential relations

In the following, is any of , , , for \begin \left(\frac\frac\right)^m \left (z^ f_n(z)\right ) &= z^ f_(z), \\ \left(\frac\frac\right)^m \left (z^ f_n(z)\right ) &= (-1)^m z^ f_(z). \end


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 ...
: \begin h_n^(x) &= j_n(x) + i y_n(x), \\ h_n^(x) &= j_n(x) - i y_n(x). \end In fact, there are simple closed-form expressions for the Bessel functions of half-integer 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 a ...
s, and therefore for the spherical Bessel functions. In particular, for non-negative integers : h_n^(x) = (-i)^ \frac \sum_^n \frac \frac, 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–Bessel functions only slightly differ from spherical Bessel functions: \begin S_n(x) &= x j_n(x) = \sqrt J_(x) \\ C_n(x) &= -x y_n(x) = -\sqrt Y_(x) \\ \xi_n(x) &= x h_n^(x) = \sqrt H_^(x) = S_n(x) - iC_n(x) \\ \zeta_n(x) &= x h_n^(x) = \sqrt H_^(x) = S_n(x) + iC_n(x) \end They satisfy the differential equation x^2 \frac + \left (x^2 - n(n + 1)\right) y = 0. 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 chemistr ...
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 ...
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 g ...
(1909), the notation , is sometimes used instead of , .


Asymptotic forms

The Bessel functions have the following asymptotic forms. For small arguments 0, one obtains, when \alpha is not a negative integer: J_\alpha(z) \sim \frac \left( \frac \right)^\alpha. When is a negative integer, we have J_\alpha(z) \sim \frac \left( \frac \right)^\alpha. For the Bessel function of the second kind we have three cases: Y_\alpha(z) \sim \begin \dfrac \left( \ln \left(\dfrac \right) + \gamma \right) & \text \alpha = 0 \\ -\dfrac \left( \dfrac \right)^\alpha + \dfrac \left(\dfrac \right)^\alpha \cot(\alpha \pi) & \text \alpha \text \alpha \text, \\ -\dfrac \left( \dfrac \right)^\alpha & \text \alpha\text \end where is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
(0.5772...). For large real arguments , one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless is half-integer) because they have zeros all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of one can write an equation containing a term of order : \begin J_\alpha(z) &= \sqrt\left(\cos \left(z-\frac - \frac\right) + e^\mathcal\left(, z, ^\right)\right) && \text \left, \arg z\ < \pi, \\ Y_\alpha(z) &= \sqrt\left(\sin \left(z-\frac - \frac\right) + e^\mathcal\left(, z, ^\right)\right) && \text \left, \arg z\ < \pi. \end (For the last terms in these formulas drop out completely; see the spherical Bessel functions above.) Even though these equations are true, better approximations may be available for complex . For example, when is near the negative real line is approximated better by J_0(z) \approx \sqrt\cos \left(z+\frac\right) than by J_0(z) \approx \sqrt\cos \left(z-\frac\right). The asymptotic forms for the Hankel functions are: \begin H_\alpha^(z) &\sim \sqrte^ && \text -\pi < \arg z < 2\pi, \\ H_\alpha^(z) &\sim \sqrte^ && \text -2\pi < \arg z < \pi. \end These can be extended to other values of using equations relating and to and . It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, is not asymptotic to the average of these two asymptotic forms when is negative (because one or the other will not be correct there, depending on the used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for ''complex'' (non-real) so long as goes to infinity at a constant phase angle (using the square root having positive real part): \begin J_\alpha(z) &\sim \frac e^ && \text -\pi < \arg z < 0, \\ J_\alpha(z) &\sim \frac e^ && \text 0 < \arg z < \pi, \\ Y_\alpha(z) &\sim -i\frac e^ && \text -\pi < \arg z < 0, \\ Y_\alpha(z) &\sim i\frac e^ && \text 0 < \arg z < \pi. \end For the modified Bessel functions, Hankel developed asymptotic (large argument) expansions as well: \begin I_\alpha(z) &\sim \frac \left(1 - \frac + \frac - \frac + \cdots \right) &&\text\left, \arg z\<\frac, \\ K_\alpha(z) &\sim \sqrt e^ \left(1 + \frac + \frac + \frac + \cdots \right) &&\text\left, \arg z\<\frac. \end There is also the asymptotic form (for large real z) \begin I_\alpha(z) = \frac\exp\left(-\alpha \operatorname\left(\frac\right) + z\sqrt\right)\left(1 + \mathcal\left(\frac\right)\right). \end When , all the terms except the first vanish, and we have \begin I_(z) &= \sqrt\sinh(z) \sim \frac && \text\left, \arg z\<\tfrac, \\ K_(z) &= \sqrt e^. \end For small arguments 0<, z, \ll\sqrt, we have \begin I_\alpha(z) &\sim \frac \left( \frac \right)^\alpha, \\ K_\alpha(z) &\sim \begin -\ln \left (\dfrac \right ) - \gamma & \text \alpha=0 \\ \frac \left( \dfrac \right)^\alpha & \text \alpha > 0 \end \end


Properties

For integer order , is often defined via a Laurent series for a generating function: e^ = \sum_^\infty J_n(x) t^n an approach used by
P. A. Hansen Peter Andreas Hansen (born 8 December 1795, Tønder, Schleswig, Denmark; died 28 March 1874, Gotha, Thuringia, Germany) was a Danish-born German astronomer. Biography The son of a goldsmith, Hansen learned the trade of a watchmaker at Flensburg, ...
in 1843. (This can be generalized to non-integer order by
contour integration In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
or other methods.) A series expansion using Bessel functions (
Kapteyn series Kapteyn series is a series expansion of analytic functions on a domain in terms of the Bessel function of the first kind. Kapteyn series are named after Willem Kapteyn, who first studied such series in 1893.Kapteyn, W. (1893). Recherches sur les fu ...
) is : \frac = 1 + 2 \sum _^ J_(nz). Another important relation for integer orders is the ''
Jacobi–Anger expansion In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindr ...
'': e^ = \sum_^\infty i^n J_n(z) e^ and e^ = J_0(z)+2\sum_^\infty J_(z) \cos(2n\phi) \pm 2i \sum_^\infty J_(z)\sin((2n+1)\phi) which is used to expand a
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...
as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal. More generally, a series f(z)=a_0^\nu J_\nu (z)+ 2 \cdot \sum_^\infty a_k^\nu J_(z) is called Neumann expansion of . The coefficients for have the explicit form a_k^0=\frac \int_ f(z) O_k(z) \,dz where is Neumann's polynomial. Selected functions admit the special representation f(z)=\sum_^\infty a_k^\nu J_(z) with a_k^\nu=2(\nu+2k) \int_0^\infty f(z) \fracz \,dz due to the orthogonality relation \int_0^\infty J_\alpha(z) J_\beta(z) \frac z= \frac 2 \pi \frac More generally, if has a branch-point near the origin of such a nature that f(z)= \sum_ a_k J_(z) then \mathcal\left\(s)=\frac\sum_\frac or \sum_ a_k \xi^= \frac \mathcal\ \left( \frac \right) where \mathcal\ is the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
of . Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula: \begin J_\nu(z) &= \frac \int_^1 e^\left(1-s^2\right)^ \,ds \\ px&=\frac 2 \int_1^\infty \frac \,du \end where and . This formula is useful especially when working with Fourier transforms. Because Bessel's equation becomes
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
(self-adjoint) if it is divided by , the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: \int_0^1 x J_\alpha\left(x u_\right) J_\alpha\left(x u_\right) \,dx = \frac \left _ \left(u_\right)\right2 = \frac \left _'\left(u_\right)\right2 where , is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
, and is the th
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
of . This orthogonality relation can then be used to extract the coefficients in the
Fourier–Bessel series 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. Fourier–Bessel series are used in the solution to partial differential e ...
, where a function is expanded in the basis of the functions for fixed and varying . An analogous relationship for the spherical Bessel functions follows immediately: \int_0^1 x^2 j_\alpha\left(x u_\right) j_\alpha\left(x u_\right) \,dx = \frac \left _\left(u_\right)\right2 If one defines a boxcar function of that depends on a small parameter as: f_\varepsilon(x)=\varepsilon \operatorname\left(\frac\varepsilon\right) (where is the
rectangle function The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname(t) = \Pi(t) = \left\{\begin{array}{r ...
) then the
Hankel transform 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 . The Bessel functions in the sum are all of the same order ν, but differ in a scaling ...
of it (of any given order ), , approaches as approaches zero, for any given . Conversely, the Hankel transform (of the same order) of is : \int_0^\infty k J_\alpha(kx) g_\varepsilon(k) \,dk = f_\varepsilon(x) which is zero everywhere except near 1. As approaches zero, the right-hand side approaches , where is the Dirac delta function. This admits the limit (in the distributional sense): \int_0^\infty k J_\alpha(kx) J_\alpha(k) \,dk = \delta(x-1) A change of variables then yields the ''closure equation'': \int_0^\infty x J_\alpha(ux) J_\alpha(vx) \,dx = \frac \delta(u - v) for . The Hankel transform can express a fairly arbitrary function as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is: \int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) \,dx = \frac \delta(u - v) for . Another important property of Bessel's equations, which follows from
Abel's identity In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a c ...
, involves the
Wronskian In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian o ...
of the solutions: A_\alpha(x) \frac - \frac B_\alpha(x) = \frac where and are any two solutions of Bessel's equation, and is a constant independent of (which depends on α and on the particular Bessel functions considered). In particular, J_\alpha(x) \frac - \frac Y_\alpha(x) = \frac and I_\alpha(x) \frac - \frac K_\alpha(x) = -\frac, for . For , the even entire function of genus 1, , has only real zeros. Let 0 be all its positive zeros, then J_(z)=\frac\prod_^\left(1-\frac\right) (There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)


Recurrence relations

The functions , , , and all satisfy the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s \frac Z_\alpha(x) = Z_(x) + Z_(x) and 2\frac = Z_(x) - Z_(x), where denotes , , , or . These two identities are often combined, e.g. added or subtracted, to yield various other relations. In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives). In particular, it follows that \begin \left( \frac \frac \right)^m \left x^\alpha Z_\alpha (x) \right&= x^ Z_ (x), \\ \left( \frac \frac \right)^m \left \frac \right&= (-1)^m \frac. \end ''Modified'' Bessel functions follow similar relations: e^ = \sum_^\infty I_n(x) t^n and e^ = I_0(z) + 2\sum_^\infty I_n(z) \cos n\theta and \frac \int_0^ e^ d\theta = I_0(z)I_0(y) + 2\sum_^\infty I_n(z)I_(y). The recurrence relation reads \begin C_(x) - C_(x) &= \frac C_\alpha(x), \\ C_(x) + C_(x) &= 2\frac, \end where denotes or . These recurrence relations are useful for discrete diffusion problems.


Transcendence

In 1929,
Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
proved that , , and the quotient are
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
s when ''ν'' is rational and ''x'' is algebraic and nonzero. The same proof also implies that is transcendental under the same assumptions.


Multiplication theorem

The Bessel functions obey a
multiplication theorem Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...
\lambda^ J_\nu(\lambda z) = \sum_^\infty \frac \left(\frac\right)^n J_(z), where and may be taken as arbitrary complex numbers.Abramowitz and Stegun
p. 363, 9.1.74
For , the above expression also holds if is replaced by . The analogous identities for modified Bessel functions and are \lambda^ I_\nu(\lambda z) = \sum_^\infty \frac \left(\frac\right)^n I_(z) and \lambda^ K_\nu(\lambda z) = \sum_^\infty \frac \left(\frac\right)^n K_(z).


Zeros of the Bessel function


Bourget's hypothesis

Bessel himself originally proved that for nonnegative integers , the equation has an infinite number of solutions in . When the functions are plotted on the same graph, though, none of the zeros seem to coincide for different values of except for the zero at . This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers and , the functions and have no common zeros other than the one at . The hypothesis was proved by
Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
in 1929.


Transcendence

Siegel proved in 1929 that when ''ν'' is rational, all nonzero roots of and are transcendental, as are all the roots of . It is also known that all roots of the higher derivatives J_\nu^(x) for ''n'' ≤ 18 are transcendental, except for the special values J_1^(\pm\sqrt3) = 0 and J_0^(\pm\sqrt3) = 0.


Numerical approaches

For numerical studies about the zeros of the Bessel function, see , and .


Numerical values

The first zero in J0 (i.e, j0,1, j0,2 and j0,3) occurs at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.Abramowitz & Stegun, p409


See also

*
Anger function In mathematics, the Anger function, introduced by , is a function defined as : \mathbf_\nu(z)=\frac \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta and is closely related to Bessel functions. The Weber function (also known as Lommel–Weber f ...
* Bessel polynomials * Bessel–Clifford function * Bessel–Maitland function *
Fourier–Bessel series 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. Fourier–Bessel series are used in the solution to partial differential e ...
* Hahn–Exton -Bessel function *
Hankel transform 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 . The Bessel functions in the sum are all of the same order ν, but differ in a scaling ...
* Incomplete Bessel functions * Jackson -Bessel function *
Kelvin functions In applied mathematics, the Kelvin functions ber''ν''(''x'') and bei''ν''(''x'') are the real and imaginary parts, respectively, of :J_\nu \left (x e^ \right ),\, where ''x'' is real, and , is the ''ν''th order Bessel function of the first kin ...
*
Kontorovich–Lebedev transform In mathematics, the Kontorovich–Lebedev transform is an integral transform which uses a Macdonald function (modified Bessel function of the second kind) with imaginary index as its kernel. Unlike other Bessel function transforms, such as the Han ...
* Lentz's algorithm *
Lerche–Newberger sum rule The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982, finds the sum of certain infinite series involving Bessel functions ''J'α'' of the first kind. It states that if ''μ'' is any non-integer comple ...
*
Lommel function The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation: : z^2 \frac + z \frac + (z^2 - \nu^2)y = z^. Solutions are given by the Lommel functions ''s''μ,ν(''z'') and ''S' ...
*
Lommel polynomial A Lommel polynomial ''R'm'',ν(''z''), introduced by , is a polynomial in 1/''z'' giving the recurrence relation :\displaystyle J_(z) = J_\nu(z)R_(z) - J_(z)R_(z) where ''J''ν(''z'') is a Bessel function of the first kind. They are given ...
*
Neumann polynomial In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions. The first few polynomials are :O_0^(t)=\frac 1 t, :O_1^(t ...
*
Schlömilch's series Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval (0,\pi) in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived th ...
*
Sonine formula In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin Nikolay Yakovlevich Sonin (Russian: Никола́й Я́ковлевич Со́нин, February 22, 1849 – February 27, 1 ...
* Struve function *
Vibrations of a circular membrane A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Due to the ph ...
* Weber function (defined at
Anger function In mathematics, the Anger function, introduced by , is a function defined as : \mathbf_\nu(z)=\frac \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta and is closely related to Bessel functions. The Weber function (also known as Lommel–Weber f ...
)


Notes


References

* * Arfken, George B. and Hans J. Weber, ''Mathematical Methods for Physicists'', 6th edition (Harcourt: San Diego, 2005). . * Bowman, Frank ''Introduction to Bessel Functions'' (Dover: New York, 1958). . * * . * . * B Spain, M. G. Smith,
Functions of mathematical physics
', Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions. * N. M. Temme, ''Special Functions. An Introduction to the Classical Functions of Mathematical Physics'', John Wiley and Sons, Inc., New York, 1996. . Chapter 9 deals with Bessel functions. * Watson, G. N., ''A Treatise on the Theory of Bessel Functions, Second Edition'', (1995) Cambridge University Press. . * . * *


External links

* . * . * . * Wolfram function pages on Besse
J
an
Y
functions, and modified Besse
I
an
K
functions. Pages include formulas, function evaluators, and plotting calculators. * * Bessel function
JνYνIν
an
Kν
in Libro
Function handbook
*F. W. J. Olver, L. C. Maximon
Bessel Functions
(chapter 10 of the Digital Library of Mathematical Functions). * {{Authority control Special hypergeometric functions Fourier analysis