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Bessel functions , first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel , are canonical solutions y x of Bessel's differential equation. Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates.
Bessel functions , first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel , are canonical solutions y x of Bessel's differential equation. 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 and static potentials.
For example:. Bessel functions also appear in other problems, such as signal processing e. Because this is a second-order linear differential equation, there must be two linearly independent 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 N n and n n , respectively, rather than Y n and y n.
On the other hand, for integer order n , 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. Another definition of the Bessel function, for integer values of n , is possible using an integral representation: .
Another integral representation is: . This was the approach that Bessel used, and from this definition he derived several properties of the function. The Bessel functions can be expressed in terms of the generalized hypergeometric series as .
This expression is related to the development of Bessel functions in terms of the Bessel—Clifford function. In terms of the Laguerre polynomials L k and arbitrarily chosen parameter t , the Bessel function can be expressed as . These are sometimes called Weber functions , as they were introduced by H.
If n is a nonnegative integer, we have the series . See also the subsection on Hankel functions below. 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. These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. 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 for the frequency.
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 as . These are chosen to be real-valued for real and positive arguments x.
Bessel functions can be described as Fourier transforms of powers of quadratic functions. It can be proven by showing equality to the above integral definition for K 0.
This is done by integrating a closed curve in the first quadrant of the complex plane. The modified Bessel function of the second kind has also been called by the following names now rare :. When solving the Helmholtz equation 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 j n and y n , and are related to the ordinary Bessel functions J n and Y n by .
The spherical Bessel functions can also be written as Rayleigh's formulas . The zeroth spherical Bessel function j 0 x is also known as the unnormalized sinc function. The first few spherical Bessel functions are: .
The spherical Bessel functions have the generating 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 :.
The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field. Riccati —Bessel functions only slightly differ from spherical Bessel functions:.
See e. The Bessel functions have the following asymptotic forms. Even though these equations are true, better approximations may be available for complex z. For example, J 0 z when z is near the negative real line is approximated better by. 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 z so long as z goes to infinity at a constant phase angle arg z using the square root having positive real part :.
For the modified Bessel functions, Hankel developed asymptotic large argument expansions as well:  . A very good approximation error below 0. Hansen in This can be generalized to non-integer order by contour integration or other methods. Another important relation for integer orders is the Jacobi—Anger expansion :. More generally, if f has a branch-point near the origin of such a nature that. Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:.
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:. This admits the limit in the distributional sense :. A change of variables then yields the closure equation : . The Hankel transform can express a fairly arbitrary function [ clarification needed ] as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:.
Another important property of Bessel's equations, which follows from Abel's identity , involves the Wronskian of the solutions:. In particular,. There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references. These two identities are often combined, e. 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 . These recurrence relations are useful for discrete diffusion problems. The Bessel functions obey a multiplication theorem. This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. The hypothesis was proved by Carl Ludwig Siegel in From Wikipedia, the free encyclopedia.
Families of solutions to related differential equations. New York: Wiley. Archived from the original on Retrieved Retrieved 25 March Accessed on line Oct. Quantum Electrodynamics. Journal of Experimental and Theoretical Physics. Bibcode : JETP Derived from formulas sourced to I. Gradshteyn and I. The Annals of Mathematical Statistics. Introduction to Quantum Mechanics, 2nd edition, p. Applied Optics.
Bibcode : ApOpt.. A Treatise on the Theory of Bessel Functions. Cambridge University Press. Retrieved 25 March — via Google Books. Table of Integrals, Series, and Products.
Translated by Scripta Technica, Inc. Academic Press, Inc. Proceedings of the National Academy of Sciences. Bibcode : PNAS Abramowitz, Milton ; Stegun, Irene Ann , eds. Applied Mathematics Series. Washington D. See also chapter Arfken, George B.
Some novel interlacing properties of the zeros for the Bessel and spherical Bessel functions are first presented and then applied to prove an interesting uniqueness result in inverse acoustic obstacle scattering. Most users should sign in with their email address. If you originally registered with a username please use that to sign in. To purchase short term access, please sign in to your Oxford Academic account above. Don't already have an Oxford Academic account? Oxford University Press is a department of the University of Oxford.
Math Lecture Notes: Bessel Functions and their Applications to. Solutions of Partial Differential Equations. Vladimir Zakharov. June 3,
Bessel functions and their applications
We introduce fractional-order Bessel functions FBFs to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived.
Three new families of Bessel function distributions are introduced by taking products of a Bessel function pdf of the first kind and a Bessel function pdf of the second kind. Various particular cases and expressions for moments are derived for each of these three distributions. Keywords Appell functions Bessel function distribution of the first kind Bessel function distribution of the second kind Elliptic integrals Hypergeometric functions Legendre functions Modified Bessel functions Probability density functions pdf Pochhammer symbol. About this region Search Form Enter search terms. Date limit:. Enter the date in the correct format.
Bessel functions are associated with a wide range of problems in important areas of mathematical physics. Bessel function theory is applied to.
Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Korenev Published Mathematics. Properties of Bessel Functions.
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