Bessel function

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:

$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0$

for an arbitrary real or complex number α (the order of the Bessel function). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A mathematician is a person whose primary area of study and research is the field of Mathematics. Daniel Bernoulli ( Groningen, 29 January 1700 &ndash 27 July 1782 was a Dutch - Swiss Mathematician, who is particularly remembered for his applications Friedrich Wilhelm Bessel (22 July 1784 &ndash 17 March 1846 was a German Mathematician, Astronomer, and systematizer of the Bessel functions Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the The most common and important special case is where α is an integer n. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

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). Bessel functions are also known as Cylinder functions or Cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates. In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually

1 Applications
2 Definitions
3 Asymptotic forms
4 Properties
5 Multiplication theorem
6 See also
7 References
8 External links

Applications

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties The Helmholtz equation, named for Hermann von Helmholtz, is the Elliptic partial differential equation (\nabla^2 + k^2 A = 0 The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. Wave propagation is any of the ways in which waves travel through a Waveguide. 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:

electromagnetic waves in a cylindrical waveguide
heat conduction in a cylindrical object. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. A waveguide is a structure which guides waves such as Electromagnetic waves Light, or Sound waves Heat conduction or thermal conduction is the spontaneous transfer of thermal energy through matter from a region of higher Temperature to a region of lower
modes of vibration of a thin circular (or annular) artificial membrane (such as a drum or other membranophone). An artificial membrane, also called a synthetic membrane, is a membrane prepared for separation tasks in Laboratory and industry The drum is a member of the percussion group technically classified as a Membranophone. A membranophone is any Musical instrument which produces Sound primarily by way of a vibrating stretched 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). A 220 Hz carrier tone modulated by a 440 Hz modulating tone with The Kaiser window is a Window function used for Digital signal processing, and is defined by the formula: w_n = \left\{ 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. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below.

Bessel functions of the first kind : $J α$

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. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical For integer order solutions, it is possible to define the function by its Taylor series expansion around x = 0:

$J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma(m+\alpha+1)} {\left({\frac{x}{2}}\right)}^{2m+\alpha}$

where $Γ(z)$ is the gamma function, a generalization of the factorial function to non-integer values. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function Definition The factorial function is formally defined by n!=\prod_{k=1}^n k For non-integer α, a more general power series expansion is required. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + 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 $- J 1 (x)$ is the derivative of $J 0 (x)$ , much like $- sin(x)$ is the derivative of $cos(x)$ ; more generally, the derivative of $J n (x)$ can be expressed in terms of $J_{n\pm 1}(x)$ by the identities below. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical )

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:

$J_{-n}(x) = (-1)^n J_{n}(x).\,$

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 $n$ , is possible using an integral representation:

$J_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \cos (n \tau - x \sin \tau) d\tau.$

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

Another integral representation is:

$J_n (x) = \frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{-i(n \tau - x \sin \tau)} d\tau$

Relation to hypergeometric series

The Bessel functions can be expressed in terms of the hypergeometric series as

$J_\alpha(z)=\frac{(z/2)^\alpha}{\Gamma(\alpha+1)} \;_0F_1 (\alpha+1; -z^2/4).$

This expression is related to the development of Bessel functions in terms of the Bessel-Clifford function. In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function In mathematical analysis the Bessel-Clifford function is an Entire function of two Complex variables which can be used to provide an alternative development of the

Bessel functions of the second kind : $Y α$

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 (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

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:

$Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}.$

In the case of integer order n, the function is defined by taking the limit as a non-integer α tends to n':

$Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x),$

which has the result (in integral form)

$Y_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \sin(x \sin\theta - n\theta)d\theta - \frac{1}{\pi} \int_{0}^{\infty} \left[ e^{n t} + (-1)^n e^{-n t} \right] e^{-x \sinh t} dt.$

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:

$Y_{-n}(x) = (-1)^n Y_n(x).\,$

Both J_α(x) and Y_α(x) are holomorphic functions of x on the complex plane cut along the negative real axis. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis When α is an integer, there is no branch point, and the Bessel functions are entire functions of x. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued In Complex analysis, an entire function, also called an integral function is a complex-valued function that is holomorphic everywhere on the If x is held fixed, then the Bessel functions are entire functions of α.

Hankel functions : $H α$

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:

$H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)$

$H_\alpha^{(2)}(x) = J_\alpha(x) - i Y_\alpha(x)$

where i is the imaginary unit. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation 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). 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 measure of the number of occurrences of a repeating event per unit Time. They are named after Hermann Hankel. Hermann Hankel ( February 14, 1839 - August 29, 1873) was a German Mathematician who was born in Halle,

Using the previous relationships they can be expressed as:

$H_{\alpha}^{(1)} (x) = \frac{J_{-\alpha} (x) - e^{-\alpha \pi i} J_\alpha (x)}{i \sin (\alpha \pi)}$

$H_{\alpha}^{(2)} (x) = \frac{J_{-\alpha} (x) - e^{\alpha \pi i} J_\alpha (x)}{- i \sin (\alpha \pi)}$

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

$H_{-\alpha}^{(1)} (x)= e^{\alpha \pi i} H_{\alpha}^{(1)} (x)$

$H_{-\alpha}^{(2)} (x)= e^{-\alpha \pi i} H_{\alpha}^{(2)} (x)$

Modified Bessel functions : $I α, K α$

The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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:

$I_\alpha(x) = i^{-\alpha} J_\alpha(ix) \!$

$K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)} = \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix) \!$

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:

$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0.$

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. Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value 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.

Modified Bessel functions of 1st kind, I_α(x), for α=0,1,2,3

Modified Bessel functions of 2nd kind, K_α(x), for α=0,1,2,3

The modified Bessel function of the second kind has also been called by the now-rare names:

Basset function
modified Bessel function of the third kind
MacDonald function

Spherical Bessel functions : $j n, y n$

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:

$x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0.$

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:

$j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x),$

$y_n(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-1/2}(x).$

$y n$ is also denoted $n n$ or η_n; some authors call these functions the spherical Neumann functions. The Helmholtz equation, named for Hermann von Helmholtz, is the Elliptic partial differential equation (\nabla^2 + k^2 A = 0 Eta (uppercase &Eta, lowercase η Ήτα) is the seventh letter of the Greek alphabet.

The spherical Bessel functions can also be written as:

$j_n(x) = (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\sin x}{x} ,$

$y_n(x) = -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\cos x}{x}.$

The first spherical Bessel function $j 0 (x)$ is also known as the (unnormalized) sinc function. In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes The first few spherical Bessel functions are:

$j_0(x)=\frac{\sin x} {x}$

$j_1(x)=\frac{\sin x} {x^2}- \frac{\cos x} {x}$

$j_2(x)=\left(\frac{3} {x^2} - 1 \right)\frac{\sin x}{x} - \frac{3\cos x} {x^2}$

and

$y_0(x)=-j_{-1}(x)=-\,\frac{\cos x} {x}$

$y_1(x)=j_{-2}(x)=-\,\frac{\cos x} {x^2}- \frac{\sin x} {x}$

$y_2(x)=-j_{-3}(x)=\left(-\,\frac{3}{x^2}+1 \right)\frac{\cos x}{x}- \frac{3 \sin x} {x^2}.$

There are also spherical analogues of the Hankel functions:

$h_n^{(1)}(x) = j_n(x) + i y_n(x)$

$h_n^{(2)}(x) = j_n(x) - i y_n(x).$

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 Mathematics, a half-integer is a Number of the form n + 1/2 where n is an Integer. In particular, for non-negative integers n:

$h_n^{(1)}(x) = (-i)^{n+1} \frac{e^{ix}}{x} \sum_{m=0}^n \frac{i^m}{m!(2x)^m} \frac{(n+m)!}{(n-m)!}$

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 : $S n, C n,ζ n$

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

$S_n(x)=x j_n(x)=\sqrt{\pi x/2}J_{n+1/2}(x)$

$C_n(x)=-x y_n(x)=-\sqrt{\pi x/2}Y_{n+1/2}(x)$

$\zeta_n(x)=x h_n^{(2)}(x)=\sqrt{\pi x/2}H_{n+1/2}^{(2)}(x)=S_n(x)+iC_n(x)$

They satisfy the differential equation:

$x^2 \frac{d^2 y}{dx^2} + [x^2 - n (n+1)] y = 0$

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). Mie theory, also called Lorenz-Mie theory or Lorenz-Mie-Debye theory, is a complete analytical solution of Maxwell's equations for the Scattering See e. g. Du (2004)^[1] for recent developments and references.

Following Debye (1909), the notation $ψ n,χ n$ is sometimes used instead of $S n, C n$ . Peter Joseph William Debye ( March 24 1884 &ndash November 2 1966) was a Dutch physicist and physical chemist

Asymptotic forms

The Bessel functions have the following asymptotic forms for non-negative α. In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which For small arguments $0 < x \ll \sqrt{\alpha + 1}$ , one obtains:

$J_\alpha(x) \rightarrow \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha$

$Y_\alpha(x) \rightarrow \left\{ \begin{matrix} \frac{2}{\pi} \left[ \ln (x/2) + \gamma \right] & \mbox{if } \alpha=0 \\ \\ -\frac{\Gamma(\alpha)}{\pi} \left( \frac{2}{x} \right) ^\alpha & \mbox{if } \alpha > 0 \end{matrix} \right.$

where γ is the Euler-Mascheroni constant (0. The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually 5772. . . ) and Γ denotes the gamma function. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function For large arguments $x \gg |\alpha^2 - 1/4|$ , they become:

$J_\alpha(x) \rightarrow \sqrt{\frac{2}{\pi x}} \cos \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right)$

$Y_\alpha(x) \rightarrow \sqrt{\frac{2}{\pi x}} \sin \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right).$

(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 $x \gg |\alpha^2 - 1/4|$ , the modified Bessel functions become:

$I_\alpha(x) \rightarrow \frac{1}{\sqrt{2\pi x}} e^x,$

$K_\alpha(x) \rightarrow \sqrt{\frac{\pi}{2x}} e^{-x}.$

while for small arguments $0 < x \ll \sqrt{\alpha + 1}$ , they become:

$I_\alpha(x) \rightarrow \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha$

$K_\alpha(x) \rightarrow \left\{ \begin{matrix} - \ln (x/2) - \gamma & \mbox{if } \alpha=0 \\ \\ \frac{\Gamma(\alpha)}{2} \left( \frac{2}{x} \right) ^\alpha & \mbox{if } \alpha > 0 \end{matrix} \right.$

Properties

For integer order α = n, J_n is often defined via a Laurent series for a generating function:

$e^{(x/2)(t-1/t)} = \sum_{n=-\infty}^\infty J_n(x) t^n,$

an approach used by P. A. Hansen in 1843. In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms Peter Andreas Hansen ( December 8, 1795 – March 28, 1874) was a Danish Astronomer, was born at Tønder (This can be generalized to non-integer order by contour integration or other methods. In complex analysis contour integration is a method of evaluating certain Integrals along paths in the complex plane ) Another important relation for integer orders is the Jacobi-Anger identity:

$e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi},$

which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone modulated FM signal. The Jacobi-Anger identity is an expansion of exponentials of Trigonometric functions in the basis of their harmonics In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

The functions J_α, Y_α, H_α⁽¹⁾, and H_α⁽²⁾ all satisfy the recurrence relations:

$\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)$

$2\frac{dZ_\alpha}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x)$

where Z denotes J, Y, H⁽¹⁾, or H⁽²⁾. "Difference equation" redirects here It should not be confused with a Differential equation. (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:

$\left( \frac{d}{x dx} \right)^m \left[ x^\alpha Z_{\alpha} (x) \right] = x^{\alpha - m} Z_{\alpha - m} (x)$

$\left( \frac{d}{x dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] = (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}.$

Modified Bessel functions follow similar relations :

$e^{(x/2)(t+1/t)} = \sum_{n=-\infty}^\infty I_n(x) t^n,$

and

$e^{z \cos \theta} = I_0(z) + 2\sum_{n=1}^\infty I_n(z) \cos(n\theta),$

The recurrence relation reads

$C_{\alpha-1}(x) + C_{\alpha+1}(x) = \frac{2\alpha}{x} C_\alpha(x)$

$C_{\alpha-1}(x) - C_{\alpha+1}(x) = 2\frac{dC_\alpha}{dx}$

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. A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint In particular, it follows that:

$\int_0^1 x J_\alpha(x u_{\alpha,m}) J_\alpha(x u_{\alpha,n}) dx = \frac{\delta_{m,n}}{2} [J_{\alpha+1}(u_{\alpha,m})]^2 = \frac{\delta_{m,n}}{2} [J_{\alpha}'(u_{\alpha,m})]^2,$

where α > -1, δ_m,n is the Kronecker delta, and u_α,m is the m-th zero of J_α(x). In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two This article is about the zeros of a function which should not be confused with the value at zero. 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. In Mathematics, Fourier–Bessel series are a particular kind of Infinite series expansion on a finite interval based on Bessel functions and as such are (An analogous relationship for the spherical Bessel functions follows immediately. )

Another orthogonality relation is the closure equation:

$\int_0^\infty x J_\alpha(ux) J_\alpha(vx) dx = \frac{1}{u} \delta(u - v)$

for α > -1/2 and where δ is the Dirac delta function. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. This property is used to construct an arbitrary function from a series of Bessel functions by means of the Hankel transform. Besides the meaning discussed in this article the Hankel transform may also refer to the determinant of the Hankel matrix of a sequence. For the spherical Bessel functions the orthogonality relation is:

$\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) dx = \frac{\pi}{2u^2} \delta(u - v)$

for α > 0.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:

$A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x},$

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). "Abel's formula" redirects here For the formula on difference operators see Summation by parts. In Mathematics, the Wronskian is a function named after the Polish mathematician Józef Hoene-Wroński. 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

$\lambda^{-\nu} J_\nu (\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{(1-\lambda^2)z}{2}\right)^n J_{\nu+n}(z)$

where $λ$ and $ν$ may be taken as arbitrary complex numbers. In Mathematics, the multiplication theorem is a certain type of identity obeyed by many Special functions related to the Gamma function. A similar form may be given for $Y ν (z)$ and etc. See ^[2]

References

^ Hong Du, "Mie-scattering calculation," Applied Optics 43 (9), 1951-1956 (2004)
^ Abramowitz, Milton & Stegun, Irene A. In mathematical analysis the Bessel-Clifford function is an Entire function of two Complex variables which can be used to provide an alternative development of the 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 Besides the meaning discussed in this article the Hankel transform may also refer to the determinant of the Hankel matrix of a sequence. In Mathematics, Fourier–Bessel series are a particular kind of Infinite series expansion on a finite interval based on Bessel functions and as such are In Mathematics, Struve functions \mathbf{H}_\alpha(x are solutions y ( x) of the non-homogenous Bessel's differential equation: , eds. (1965), “Chapter 9”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 . Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U

Abramowitz, Milton & Stegun, Irene A. , eds. (1965), “Chapter 9”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 See also chapter 10. Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U
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.

External links

Lizorkin, P. I. (2001), “Bessel functions”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
L. The Encyclopaedia of Mathematics is a large reference work in Mathematics. N. Karmazina (2001), “Cylinder function”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
N. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Kh. Rozov (2001), “Bessel equation”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Bessel functions of order ν (Javascript)
Wolfram Mathworld - Bessel functions of the first kind
Bessel functions applied to Acoustic Field analysis on Trinnov Audio's research page

The Encyclopaedia of Mathematics is a large reference work in Mathematics.

Dictionary

-noun

(mathematics) Any of a class of functions that are solutions to a particular form of differential equation (a Bessel equation) and are typically used to describe waves in a cylindrically symmetric system.

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