In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Daniel Bernoulli Daniel Bernoulli (February 8, 1700 – March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland where he died. ... Friedrich Wilhelm Bessel (July 22, 1784 – March 17, 1846) was a German mathematician, astronomer, and systematizer of the Bessel functions (which, despite their name, were discovered by Daniel Bernoulli). ... Canonical is an adjective derived from canon. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

x²

for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred to as the order of the Bessel function. The integers are commonly denoted by the above symbol. ...

Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of α).

Applications

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n+½). For example: In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Wave propagation refers to the ways waves travel through a medium (waveguide). ...

• electromagnetic waves in a cylindrical waveguide
• heat conduction in a cylindrical object.
• modes of vibration of a thin circular (or annular) artificial membrane.
• diffusion problems on a lattice.

Bessel functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter). Electromagnetic radiation or EM radiation is a combination (cross product) of oscillating electric and magnetic fields perpendicular to each other, moving through space as a wave, effectively transporting energy and momentum. ... Look up waveguide in Wiktionary, the free dictionary. ... Heat flow along perfectly insulated wire Conduction is the transfer of heat or electric current from one substance to another by direct contact. ... An artificial membrane, also called a synthetic membrane, is a membrane prepared for separation tasks in laboratory and industry. ... Frequency modulation synthesis (or FM synthesis) is a form of audio synthesis where the timbre of a simple waveform is changed by frequency modulating it with a modulating frequency that is also in the audio range, resulting in a more complex waveform and a different-sounding tone. ... The Kaiser window is a nearly optimal window function wk used for digital signal processing, and is defined by the formula: Kaiser window function for n=100 and α= 0. ... In electronics and signal processing, a Bessel filter is a variety of linear filter with a maximally flat group delay (linear phase response). ...

Definitions

Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below. In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

Bessel functions of the first kind

Bessel functions of the first kind, denoted as J_α(x), are solutions of Bessel's differential equation that are finite at the origin (x = 0) for non-negative integer α, and diverge as x approaches zero for negative non-integer α. The solution type (e.g. integer or non-integer) and normalization of J_α(x) are defined by its properties below. For integer order solutions, it is possible to define the function by its Taylor series expansion around x = 0: In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred to... As the degree of the Taylor series rises, it approaches the correct function. ...

where Γ(z) is the gamma function, a generalization of the factorial function to non-integer values. For non-integer α, a more general power series expansion is required. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√x (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large x. (The Taylor series indicates that − J₁(x) is the derivative of J₀(x), much like − sin is the derivative of cos; more generally, the derivative of J_n(x) can be expressed in terms of by the identities below.) The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... For factorial rings in mathematics, see unique factorisation domain. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred to...

Plot of Bessel function of the first kind, J_α(x), for integer orders α=0,1,2.

For non-integer α, the functions J_α(x) and J _{− α}(x) are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order α, the following relationship is valid: Image File history File links BesselJ_plot. ... Image File history File links BesselJ_plot. ...

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 n, is possible using an integral representation:

This was the approach that Bessel used, and from this definition he derived several properties of the function.

Another integral representation is:

Relation to hypergeometric series

The Bessel functions can be expressed in terms of the hypergeometric series as In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...

This expression is related to the development of Bessel functions in terms of the Bessel-Clifford function. In mathematical analysis, the Bessel-Clifford function is a an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. ...

Bessel functions of the second kind

The Bessel functions of the second kind, denoted by Y_α(x), are solutions of the Bessel differential equation. They are singular (infinite) at the origin (x = 0). Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...

Plot of Bessel function of the second kind, Y_α(x), for integer orders α=0,1,2.

Y_α(x) is sometimes also called the Neumann function, and is occasionally denoted instead by N_α(x). For non-integer α, it is related to J_α(x) by: Image File history File links BesselY_plot. ... Image File history File links BesselY_plot. ...

In the case of integer order α, the relation is defined by taking the limit of α from non-integer to integer order:

which has the result (in integral form)

For the case of non-integer α, the definition of Y_α(x) is redundant (as is clear from its definition above). On the other hand, when α is an integer, Y_α(x) is the second linearly independent solution of Bessel's equation; moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

Both J_α(x) and Y_α(x) are holomorphic functions of x on the complex plane cut along the negative real axis. When α is an integer, there is no branch point, and the Bessel functions are entire functions of x. If x is held fixed, then the Bessel functions are entire functions of α. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ...

Hankel functions

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions H_α⁽¹⁾(x) and H_α⁽²⁾(x), defined by:

where i is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. The Hankel functions of the first and second kind 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 for the frequency). They are named after Hermann Hankel. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In physics, a sign convention is a choice of the signs (plus or minus) of a set of quantities, in a case where the choice of sign is arbitrary. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... Hermann Hankel (February 14, 1839 - August 29, 1873) was a German mathematician who was born in Halle, Germany and died in Schramberg (near Tübingen), Germany. ...

Using the previous relationships they can be expressed as:

if α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:

Modified Bessel functions

The Bessel functions are valid even for complex arguments x, 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 by: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

These are chosen to be real-valued for real arguments x. The series expansion for I_α(x) is thus similar to that for J_α(x), but without the alternating (-1)^m factor.

I_α(x) and K_α(x) 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, I_α and K_α are exponentially growing and decaying functions, respectively. Like the ordinary Bessel function J_α, the function I_α goes to zero at x=0 for α > 0 and is finite at x=0 for α=0. Analogously, K_α diverges at x=0. In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

Modified Bessel functions of 1st kind, Iα(x), for α=0,1,2

Modified Bessel functions of 2nd kind, Kα(x), for α=0,1,2

The modified Bessel function of the second kind has also been called by the now-rare names: Image File history File links BesselI_plot. ... Image File history File links BesselI_plot. ... Image File history File links BesselK_plot. ... Image File history File links BesselK_plot. ...

• Basset function
• modified Bessel function of the third kind
• MacDonald function

Spherical Bessel functions

Spherical Bessel functions of 1st kind, j_n(x), for n=0,1,2

Spherical Bessel functions of 2nd kind, y_n(x), for n=0,1,2

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form: Image File history File links Spherical_bessel_j_plot. ... Image File history File links Spherical_bessel_j_plot. ... Image File history File links Spherical_bessel_y_plot. ... Image File history File links Spherical_bessel_y_plot. ... The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ...

The two linearly independent solutions to this equation are called the spherical Bessel functions j_n and y_n, and are related to the ordinary Bessel functions J_n and Y_n by:

y_n is also denoted n_n or η_n; some authors call these functions the spherical Neumann functions. Look up Η, η in Wiktionary, the free dictionary. ...

The spherical Bessel functions can also be written as:

The first spherical Bessel function j₀(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ...

and

There are also spherical analogues of the Hankel functions:

In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers n: In mathematics, a half-integer is a number of the form , where is an integer. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...

and h_n⁽²⁾ is the complex-conjugate of this (for real x). It follows, for example, that j₀(x) = sin(x)/x and y₀(x) = -cos(x)/x, and so on.

Riccati-Bessel functions

Riccati-Bessel functions only slightly differ from spherical Bessel functions:

They satisfy the differential equation:

This differential equation, and the Riccati-Bessel solutions, arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g. Du (2004)^[1] for recent developments and references. The Mie theory also called Lorenz-Mie theory is a complete mathematical-physical theory of the scattering of electromagnetic radiation by spherical particles, developed by Gustav Mie in 1908. ...

Following Debye (1909), the notation ψ_n,χ_n is sometimes used instead of S_n,C_n. Petrus Josephus Wilhelmus Debije (March 24, 1884 – November 2, 1966) was a Dutch physical chemist. ...

Asymptotic forms

The Bessel functions have the following asymptotic forms for non-negative α. For small arguments , one obtains: 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 a given function as the argument of the function tends towards a particular...

where γ is the Euler-Mascheroni constant (0.5772...) and Γ denotes the gamma function. For large arguments , they become: The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...

(For α=1/2 these formulas are exact; see the spherical Bessel functions above.) Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations. For example, for large , the modified Bessel functions become:

while for small arguments , they become:

Properties

For integer order α = n, J_n is often defined via a Laurent series for a generating function: A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...

an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.) Another important relation for integer orders is the Jacobi-Anger identity: Peter Andreas Hansen (December 8, 1795 – March 28, 1874), Danish astronomer, was born at Tondern, in the duchy of Schleswig (now Tønder, Denmark). ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

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. In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In telecommunications, frequency modulation (FM) conveys information over a carrier wave by varying its frequency. ...

The functions J_α, Y_α, H_α⁽¹⁾, and H_α⁽²⁾ all satisfy the recurrence relations: In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

where Z denotes J, Y, H⁽¹⁾, or H⁽²⁾. (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:

Modified Bessel functions follow similar relations :

and

The recurrence relation reads

where C_α denotes I_α or e^απiK_α. These recurrence relations are useful for discrete diffusion problems.

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

where α > -1, δ_m,n is the Kronecker delta, and u_α,m is the m-th zero of J_α(x). 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 J_α(x u_α,m) for fixed α and varying m. (An analogous relationship for the spherical Bessel functions follows immediately.) In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, Fourier-Bessel series are a particular kind of infinite series expansion, based on the properties of orthogonal functions. ...

Another orthogonality relation is the closure equation:

for α > -1/2 and where δ is the Dirac delta function. For the spherical Bessel functions the orthogonality relation is: The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...

for α > 0.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions: In mathematics, Abels identity (also called Abels differential equation identity) is an equation that expresses the Wronskian of two homogeneous solutions of a second-order linear ordinary differential equation in terms of the coefficients of the original differential equation. ... In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations. ...

where A_α and B_α are any two solutions of Bessel's equation, and C_α is a constant independent of x (which depends on α and on the particular Bessel functions considered). For example, if A_α = J_α and B_α = Y_α, then C_α is 2/π. This also holds for the modified Bessel functions; for example, if A_α = I_α and B_α = K_α, then C_α is -1.

(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)

Multiplication theorem

The Bessel functions obey a multiplication theorem In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. ...

where λ and ν may be taken as arbitrary complex numbers. A similar form may be given for Y_ν(z) and etc. See ^[2]

See also

• Bessel-Clifford function
• Propagator
• Hankel Transform

In mathematical analysis, the Bessel-Clifford function is a an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ... In mathematics, the Hankel transform of order ν of a function f(r) is given by: where Jν is the Bessel function of the first kind of order ν with ν â‰¥ âˆ’1/2. ...

References

1. ^ Hong Du, "Mie-scattering calculation," Applied Optics 43 (9), 1951-1956 (2004)
2. ^ Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4.  (See chapter 9)

• Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4.  (See chapter 9 and chapter 10)
• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005). ISBN 0-12-059876-0
• Frank Bowman, Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4.
• G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, (1995) Cambridge University Press. ISBN 0-521-48391-3
• G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", Ann. Phys. Leipzig 25(1908), p.377.
• Refaat El Attar, Special Functions and Orthogonal Polynomials, (2006) Lulu Press Inc. ISBN 1-4116-6690-9
• Refaat El Attar, Bessel and Related Functions, (2007) Lulu Press Inc. ISBN 1-4303-1393-5

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

External links

• Lizorkin, P.I. (2001), "Bessel functions", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 1-55608-010-7
• L.N. Karmazina (2001), "Cylinder function", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 1-55608-010-7
• N.Kh. Rozov (2001), "Bessel equation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 1-55608-010-7
• Using the Bessel function, scientists write text on water surface. [1]

The Encyclopaedia of Mathematics is large reference work in mathematics, it is avaliable in book for, on CD-ROM and online. ... The Encyclopaedia of Mathematics is large reference work in mathematics, it is avaliable in book for, on CD-ROM and online. ... The Encyclopaedia of Mathematics is large reference work in mathematics, it is avaliable in book for, on CD-ROM and online. ...