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:

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

for an arbitrary real or complex number α (the order of the Bessel function); the most common and important cases are for α an integer or half-integer.

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.

Applications of Bessel function

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. 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

Modes of vibration of a thin circular (or annular) artificial membrane (such as a drum or other membranophone)

Diffusion problems on a lattice

Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle

Solving for patterns of acoustical radiation

Bessel functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).

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.

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 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. It is possible to define the function by its Taylor series expansion around x = 0:

:$J\_\backslash alpha(x)\; =\; \backslash sum\_\{m=0\}^\backslash infty\; \backslash frac\{(-1)^m\}\{m!\; \backslash ,\; \backslash Gamma(m+\backslash alpha+1)\}\; \{\backslash left(\backslash tfrac\{1\}\{2\}x\backslash right)\}^\{2m+\backslash alpha\}$

where Γ(z) is the gamma function, a generalization of the factorial function to non-integer values. 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 x is the derivative of cos x; more generally, the derivative of J_n(x) can be expressed in terms of J_n±1(x) by the identities below.)

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 (note that the Gamma function becomes infinite for negative integer arguments):

:$J\_\{-n\}(x)\; =\; (-1)^n\; J\_\{n\}(x).\backslash ,$

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:

:$J\_n(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash pi\; \backslash cos\; (n\; \backslash tau\; -\; x\; \backslash sin\; \backslash tau)\; \backslash ,\backslash mathrm\{d\}\backslash tau.$

Another integral representation is:

:$J\_n\; (x)\; =\; \backslash frac\{1\}\{2\; \backslash pi\}\; \backslash int\_\{-\backslash pi\}^\backslash pi\; e^\{-\backslash mathrm\{i\}\backslash ,(n\; \backslash tau\; -\; x\; \backslash sin\; \backslash tau)\}\; \backslash ,\backslash mathrm\{d\}\backslash tau.$

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 the addition of another term

:$J\_\backslash alpha(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash pi\; \backslash cos(\backslash alpha\backslash tau-\; x\; \backslash sin\backslash tau)\backslash ,d\backslash tau$

- \frac{\sin(\alpha\pi)}{\pi} \int_0^\infty e^{-x \sinh(t) - \alpha t} \, dt.

or for $\backslash alpha\; >\; -\backslash frac\{1\}\{2\}$ by :$J\_\backslash alpha(x)=\; \backslash frac\{1\}\{2^\{\backslash alpha-1\}\backslash Gamma(\backslash alpha\; +\; \backslash frac\{1\}\{2\})\; \backslash sqrt\{\backslash pi\}\backslash ,\; x^\backslash alpha\}\; \backslash int\_0^x\; (x^2-\backslash tau^2)^\{\backslash alpha-1/2\}\backslash cos\; \backslash tau\; \backslash ,\; d\backslash tau.$

Relation to hypergeometric series

The Bessel functions can be expressed in terms of the generalized hypergeometric series as :$J\_\backslash alpha(x)=\backslash frac\{(x/2)^\backslash alpha\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash ;\_0F\_1\; (\backslash alpha+1;\; -\backslash tfrac\{1\}\{4\}x^2).$ 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 $L\_k$ and arbitrarily chosen parameter $t$, the Bessel function can be expressed as :$\backslash frac\{J\_\backslash alpha(x)\}\{\backslash left(\; \backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}=\; \backslash frac\{e^\{-t\}\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash sum\_\{k=0\}\; \backslash frac\{L\_k^\{(\backslash alpha)\}\backslash left(\; \backslash frac\{x^2\}\{4\; t\}\backslash right)\}\backslash frac\{t^k\}\{k!\}.$

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 have a singularity at the origin (x = 0).

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:

:$Y\_\backslash alpha(x)\; =\; \backslash frac\{J\_\backslash alpha(x)\; \backslash cos(\backslash alpha\backslash pi)\; -\; J\_\{-\backslash alpha\}(x)\}\{\backslash sin(\backslash alpha\backslash pi)\}.$

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

:$Y\_n(x)\; =\; \backslash lim\_\{\backslash alpha\; \backslash to\; n\}\; Y\_\backslash alpha(x),$

which has the result (in integral form)

:$Y\_n(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash pi\; \backslash sin(x\; \backslash sin\backslash theta\; -\; n\backslash theta)\; \backslash ,\; d\backslash theta$

- \frac{1}{\pi} \int_0^\infty \left[ e^{n t} + (-1)^n e^{-n t} \right] e^{-x \sinh t} \, dt.

Y_α(x) is necessary as the second linearly independent solution of the Bessel's equation when α is an integer. But Y_α(x) has more meaning than that. It can be considered as a 'natural' partner of J_α(x). See also the subsection on Hankel functions below.

When α is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

:$Y\_\{-n\}(x)\; =\; (-1)^n\; Y\_n(x).\backslash ,$

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, the Bessel functions J are entire functions of x. If x is held fixed, then the Bessel functions are entire functions of α.

Hankel functions: H_α⁽¹⁾, H_α⁽²⁾

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:

:$H\_\backslash alpha^\{(1)\}(x)\; =\; J\_\backslash alpha(x)\; +\; i\; Y\_\backslash alpha(x)$

:$H\_\backslash alpha^\{(2)\}(x)\; =\; J\_\backslash alpha(x)\; -\; i\; Y\_\backslash alpha(x)$

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. They are named after Hermann Hankel.

The importance of Hankel functions of the first and second kind lies more in theoretical development rather than in application. These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, 'simple' means an appearance of the factor of the form $e^\{if(x)\}$. The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.

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).

Using the previous relationships they can be expressed as:

:$H\_\backslash alpha^\{(1)\}\; (x)\; =\; \backslash frac\{J\_\{-\backslash alpha\}\; (x)\; -\; e^\{-\backslash alpha\; \backslash pi\; i\}\; J\_\backslash alpha\; (x)\}\{i\; \backslash sin\; (\backslash alpha\; \backslash pi)\}$

:$H\_\backslash alpha^\{(2)\}\; (x)\; =\; \backslash frac\{J\_\{-\backslash alpha\}\; (x)\; -\; e^\{\backslash alpha\; \backslash pi\; i\}\; J\_\backslash alpha\; (x)\}\{-\; i\; \backslash sin\; (\backslash alpha\; \backslash pi)\}$

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

:$H\_\{-\backslash alpha\}^\{(1)\}\; (x)=\; e^\{\backslash alpha\; \backslash pi\; i\}\; H\_\backslash alpha^\{(1)\}\; (x)$

:$H\_\{-\backslash alpha\}^\{(2)\}\; (x)=\; e^\{-\backslash alpha\; \backslash pi\; i\}\; H\_\backslash alpha^\{(2)\}\; (x).$

The Hankel functions admit the following integral representations (useful in the calculus of the propagator of the Klein–Gordon field):

:$H\_\backslash alpha^\{(1)\}\; (x)=\; \backslash frac\{e^\{-\backslash frac\{1\}\{2\}\; \backslash alpha\backslash pi\; i\}\}\{\backslash pi\; i\}\backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; e^\{ix\backslash cosh\; t\; -\; \backslash alpha\; t\}\; \backslash ,\; dt.$

:$H\_\backslash alpha^\{(2)\}\; (x)=\; -\backslash frac\{e^\{-\backslash frac\{1\}\{2\}\; \backslash alpha\backslash pi\; i\}\}\{\backslash pi\; i\}\backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; e^\{-ix\backslash cosh\; t\; -\; \backslash alpha\; t\}\; \backslash ,\; dt.$

=== 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. 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 any of these equivalent alternatives:

:$I\_\backslash alpha(x)\; =\; i^\{-\backslash alpha\}\; J\_\backslash alpha(ix)\; =\backslash sum\_\{m=0\}^\backslash infty\; \backslash frac\{1\}\{m!\; \backslash Gamma(m+\backslash alpha+1)\}\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\{2m+\backslash alpha\}$

:$K\_\backslash alpha(x)\; =\; \backslash frac\{\backslash pi\}\{2\}\; \backslash frac\{I\_\{-\backslash alpha\}\; (x)\; -\; I\_\backslash alpha\; (x)\}\{\backslash sin\; (\backslash alpha\; \backslash pi)\}\; =\; \backslash frac\{\backslash pi\}\{2\}\; i^\{\backslash alpha+1\}\; H\_\backslash alpha^\{(1)\}(ix)\; =\; -\backslash frac\{\backslash pi\}\{2\}\; i^\{\backslash alpha+1\}\; e^\{-i\; \backslash pi\; \backslash alpha\}\; H\_\backslash alpha^\{(2)\}(-ix).$

There exist many integral representations of these functions. The following for K_α(x), is useful for the calculus of the Feynman propagator in field theory:

:$K\_\backslash alpha(x)\; =\; \backslash frac\{1\}\{2\}\; e^\{-\backslash frac\{1\}\{2\}\backslash alpha\backslash pi\; i\}\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; e^\{-ix\backslash sinh\; t\; -\backslash alpha\; t\}\; \backslash ,\; dt$

These are chosen to be real-valued for real and positive 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:

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

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. 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.

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Modified Bessel functions $K\_\{1/3\}$ and $K\_\{2/3\}$ can be represented in terms of rapidly converged integrals

:$K\_\{1/3\}\; (\backslash xi)\; =\; \backslash sqrt\{3\}\backslash ,\; \backslash int\_0^\backslash infty\; \backslash ,\; \backslash exp\; \backslash left[-\; \backslash xi\; \backslash left(1+\backslash frac\{4x^2\}\{3\}\backslash right)\; \backslash sqrt\{1+\backslash frac\{x^2\}\{3\}\}\; \backslash right]\; \backslash \; dx$

:$K\_\{2/3\}\; (\backslash xi)\; =\; \backslash frac\{1\}\{\; \backslash sqrt\{3\}\}\; \backslash ,\; \backslash int\_0^\backslash infty\; \backslash ,\; \backslash frac\{3+2x^2\}\{\backslash sqrt\{1+x^2/3\}\}\; \backslash exp\; \backslash left[-\; \backslash xi\; \backslash left(1+\backslash frac\{4x^2\}\{3\}\backslash right)\; \backslash sqrt\{1+\backslash frac\{x^2\}\{3\}\}\; \backslash right]\; \backslash \; dx$

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

modified Hankel function

MacDonald function

Spherical Bessel functions: j_n, y_n

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form:

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

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:

:$j\_\{n\}(x)\; =\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2x\}\}\; J\_\{n+1/2\}(x),$

:$y\_\{n\}(x)\; =\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2x\}\}\; Y\_\{n+1/2\}(x)\; =\; (-1)^\{n+1\}\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2x\}\}\; J\_\{-n-1/2\}(x).$

$y\_n$ is also denoted $n\_n$ or η_n; some authors call these functions the spherical Neumann functions.

The spherical Bessel functions can also be written as (Rayleigh's Formulas): :$j\_n(x)\; =\; (-x)^n\; \backslash left(\backslash frac\{1\}\{x\}\backslash frac\{d\}\{dx\}\backslash right)^n\backslash ,\backslash frac\{\backslash sin\; x\}\{x\}\; ,$ :$y\_n(x)\; =\; -(-x)^n\; \backslash left(\backslash frac\{1\}\{x\}\backslash frac\{d\}\{dx\}\backslash right)^n\backslash ,\backslash frac\{\backslash cos\; x\}\{x\}.$

The first spherical Bessel function $j\_0(x)$ is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: :$j\_0(x)=\backslash frac\{\backslash sin\; x\}\; \{x\}$ :$j\_1(x)=\backslash frac\{\backslash sin\; x\}\; \{x^2\}-\; \backslash frac\{\backslash cos\; x\}\; \{x\}$ :$j\_2(x)=\backslash left(\backslash frac\{3\}\; \{x^2\}\; -\; 1\; \backslash right)\backslash frac\{\backslash sin\; x\}\{x\}\; -\; \backslash frac\{3\backslash cos\; x\}\; \{x^2\}$ :$j\_3(x)=\backslash left(\backslash frac\{15\}\{x^3\}\; -\; \backslash frac\{6\}\{x\}\; \backslash right)\backslash frac\{\backslash sin\; x\}\{x\}\; -\backslash left(\backslash frac\{15\}\{x^2\}\; -\; 1\backslash right)\; \backslash frac\{\backslash cos\; x\}\; \{x\},$ and :$y\_0(x)=-j\_\{-1\}(x)=-\backslash ,\backslash frac\{\backslash cos\; x\}\; \{x\}$ :$y\_1(x)=j\_\{-2\}(x)=-\backslash ,\backslash frac\{\backslash cos\; x\}\; \{x^2\}-\; \backslash frac\{\backslash sin\; x\}\; \{x\}$ :$y\_2(x)=-j\_\{-3\}(x)=\backslash left(-\backslash ,\backslash frac\{3\}\{x^2\}+1\; \backslash right)\backslash frac\{\backslash cos\; x\}\{x\}-\; \backslash frac\{3\; \backslash sin\; x\}\; \{x^2\}$ :$y\_\{3\}\backslash left(\; x\backslash right)=j\_\{-4\}(x)\; =\backslash left(\; -\backslash frac\{15\}\{x^\{3\}\}+\backslash frac\{6\}\{x\}\backslash right)\; \backslash frac\{\backslash cos\; x\}\{x\}-\backslash left(\; \backslash frac\{15\}\{x^\{2\}\}-1\backslash right)\; \backslash frac\{\backslash sin\; x\}\{x\}.$

The general identity is : where the upper limit of summation is understood to be the largest integer less than or equal to (n+1)/2.

Generating function

The spherical Bessel functions have the generating functions :$\backslash frac\; 1\; \{z\}\; \backslash cos\; \backslash sqrt\{z^2\; -\; 2zt\}=\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{t^n\}\{n!\}\; j\_\{n-1\}(z),$ :$\backslash frac\; 1\; \{z\}\; \backslash sin\; \backslash sqrt\{z^2\; +\; 2zt\}=\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-t)^n\}\{n!\}\; y\_\{n-1\}(z)\; .$

Differential relations

In the following $f\_n$ is any of $j\_n,\; y\_n,\; h\_n^\{(1)\},\; h\_n^\{(2)\}$ for $n=0,\backslash pm\; 1,\backslash pm\; 2,\backslash dots$ :$\backslash left(\backslash frac\{1\}\{z\}\backslash frac\{d\}\{dz\}\backslash right)^m\backslash left(z^\{n+1\}f\_n(z)\backslash right)=z^\{(n-m)+1\}f\_\{(n-m)\}(z).$

Spherical Hankel functions: h _n

There are also spherical analogues of the Hankel functions:

:$h\_n^\{(1)\}(x)\; =\; j\_n(x)\; +\; i\; y\_n(x)\; \backslash ,$

:$h\_n^\{(2)\}(x)\; =\; j\_n(x)\; -\; i\; y\_n(x).\; \backslash ,$

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:

:$h\_n^\{(1)\}(x)\; =\; (-i)^\{n+1\}\; \backslash frac\{e^\{ix\}\}\{x\}\; \backslash sum\_\{m=0\}^n\; \backslash frac\{i^m\}\{m!(2x)^m\}\; \backslash frac\{(n+m)!\}\{(n-m)!\}$

and $h\_n^\{(2)\}$ is the complex-conjugate of this (for real $x$). It follows, for example, that $j\_0(x)\; =\; \backslash sin(x)/x$ and $y\_0(x)\; =\; -\backslash cos(x)/x$, and so on.

Riccati–Bessel functions: S_n, C_n, ξ_n, ζ_n

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

:$S\_n(x)=x\; j\_n(x)=\backslash sqrt\{\backslash pi\; x/2\}\; \backslash ,\; J\_\{n+1/2\}(x)$

:$C\_n(x)=-x\; y\_n(x)=-\backslash sqrt\{\backslash pi\; x/2\}\; \backslash ,\; Y\_\{n+1/2\}(x)$

:$\backslash xi\_n(x)\; =\; x\; h\_n^\{(1)\}(x)=\backslash sqrt\{\backslash pi\; x/2\}\; \backslash ,\; H\_\{n+1/2\}^\{(1)\}(x)=S\_n(x)-iC\_n(x)$

:$\backslash zeta\_n(x)=x\; h\_n^\{(2)\}(x)=\backslash sqrt\{\backslash pi\; x/2\}\; \backslash ,\; H\_\{n+1/2\}^\{(2)\}(x)=S\_n(x)+iC\_n(x).$

They satisfy the differential equation:

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

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). See e.g., Du (2004) for recent developments and references.

Following Debye (1909), the notation $\backslash psi\_n,\backslash chi\_n$ is sometimes used instead of $S\_n,C\_n$.

Asymptotic forms

The Bessel functions have the following asymptotic forms for non-negative α. For small arguments , one obtains:

:$J\_\backslash alpha(x)\; \backslash approx\; \backslash frac\{1\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash left(\; \backslash frac\{x\}\{2\}\; \backslash right)\; ^\backslash alpha$

:

where $\backslash gamma$ is the Euler–Mascheroni constant (0.5772...) and $\backslash Gamma$ denotes the gamma function. For large arguments $x\; \backslash gg\; |\backslash alpha^2\; -\; 1/4|$, they become:

:$K\_\backslash alpha(x)\; \backslash approx\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2x\}\}\; e^\{-x\}\; \backslash left(1\; +\; \backslash frac\{4\; \backslash alpha^\{2\}\; -\; 1\}\{8\; x\}\; +\; \backslash frac\{(4\; \backslash alpha^\{2\}\; -\; 1)\; (4\; \backslash alpha^\{2\}\; -\; 9)\}\{2!\; (8\; x)^\{2\}\}\; +\; \backslash frac\{(4\; \backslash alpha^\{2\}\; -\; 1)\; (4\; \backslash alpha^\{2\}\; -\; 9)\; (4\; \backslash alpha^\{2\}\; -\; 25)\}\{3!\; (8\; x)^\{3\}\}\; +\; \backslash cdots\; \backslash right)\; .$

Similarly, the last expressions are exact when $\backslash alpha=1/2$.

For small arguments , they become:

:$I\_\backslash alpha(x)\; \backslash approx\; \backslash frac\{1\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash left(\; \backslash frac\{x\}\{2\}\; \backslash right)\; ^\backslash alpha$

:

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

:$e^\{(x/2)(t-1/t)\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; J\_n(x)\; t^n,$

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 expansion:

:$e^\{iz\; \backslash cos\; \backslash phi\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; i^n\; J\_n(z)\; e^\{in\backslash phi\},$

and

:$e^\{iz\; \backslash sin\; \backslash phi\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; J\_n(z)\; e^\{in\backslash phi\},$

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.

More generally, a series :$f(z)=a\_0^\backslash nu\; J\_\backslash nu\; (z)+\; 2\; \backslash cdot\; \backslash sum\_\{k=1\}\; a\_k^\backslash nu\; J\_\{\backslash nu+k\}(z)$ is called Neumann expansion of ƒ. The coefficients for $\backslash nu=0$ have the explicit form

: $a\_k^0=\backslash frac\{1\}\{2\; \backslash pi\; i\}\; \backslash int\_\{|z|=c\}\; f(z)\; O\_k(z)\; \backslash ,\; \backslash mathrm\; d\; z,$

where $O\_k$ is Neumann's polynomial.

Selected functions admit the special representation :$f(z)=\backslash sum\_\{k=0\}\; a\_k^\backslash nu\; J\_\{\backslash nu+2k\}(z)$ with :$a\_k^\backslash nu=2(\backslash nu+2k)\; \backslash int\_0^\backslash infty\; f(z)\; \backslash frac\{J\_\{\backslash nu+2k\}(z)\}z\; \backslash mathrm\; d\; z$ due to the orthogonality relation $\backslash int\_0^\backslash infty\; J\_\backslash alpha(z)\; J\_\backslash beta(z)\; \backslash frac\; \{\backslash mathrm\; d\; z\}\; z=\; \backslash frac\; 2\; \backslash pi\; \backslash frac\{\backslash sin\backslash left(\backslash frac\; \backslash pi\; 2\; (\backslash alpha-\backslash beta)\; \backslash right)\}\{\backslash alpha^2\; -\backslash beta^2\}.$

More generally, if ƒ has a branch-point near the origin of such a nature that $f(z)=\; \backslash sum\_\{k=0\}\; a\_k\; J\_\{\backslash nu+k\}(z),$ then

:$\backslash mathcal\; L\; \backslash left\backslash \{\backslash sum\_\{k=0\}\; a\_k\; J\_\{\backslash nu+k\}\; \backslash right\backslash \}(s)=\; \backslash frac\; 1\; \backslash sqrt\{1+s^2\}\; \backslash sum\_\{k=0\}\; \backslash frac\{a\_k\}\{(s+\backslash sqrt\{1+s^2\})^\{\backslash nu+k\}\}$ or

:$\backslash sum\_\{k=0\}\; a\_k\; \backslash xi^\{\backslash nu+k\}=\; \backslash frac\{1+\backslash xi^2\}\{2\backslash xi\}\; \backslash mathcal\; L\; \backslash \{f\; \backslash \}\; \backslash left(\; \backslash frac\{1-\backslash xi^2\}\{2\backslash xi\}\; \backslash right),$

where $\backslash mathcal\; L\; \backslash \{f\; \backslash \}$ is ƒ's Laplace transform.

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:

:

where ν > −1/2 and z is a complex number. This formula is useful especially when working with Fourier transforms.

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

:$\backslash frac\{2\backslash alpha\}\{x\}\; Z\_\backslash alpha(x)\; =\; Z\_\{\backslash alpha-1\}(x)\; +\; Z\_\{\backslash alpha+1\}(x)$

:$2\backslash frac\{dZ\_\backslash alpha\}\{dx\}\; =\; Z\_\{\backslash alpha-1\}(x)\; -\; Z\_\{\backslash alpha+1\}(x)$

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:

:$\backslash left(\; \backslash frac\{d\}\{x\; dx\}\; \backslash right)^m\; \backslash left[\; x^\backslash alpha\; Z\_\{\backslash alpha\}\; (x)\; \backslash right]\; =\; x^\{\backslash alpha\; -\; m\}\; Z\_\{\backslash alpha\; -\; m\}\; (x)$

:$\backslash left(\; \backslash frac\{d\}\{x\; dx\}\; \backslash right)^m\; \backslash left[\; \backslash frac\{Z\_\backslash alpha\; (x)\}\{x^\backslash alpha\}\; \backslash right]\; =\; (-1)^m\; \backslash frac\{Z\_\{\backslash alpha\; +\; m\}\; (x)\}\{x^\{\backslash alpha\; +\; m\}\}.$

Modified Bessel functions follow similar relations :

:$e^\{(x/2)(t+1/t)\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; I\_n(x)\; t^n,$

and

:$e^\{z\; \backslash cos\; \backslash theta\}\; =\; I\_0(z)\; +\; 2\backslash sum\_\{n=1\}^\backslash infty\; I\_n(z)\; \backslash cos(n\backslash theta),$

The recurrence relation reads

:$C\_\{\backslash alpha-1\}(x)\; -\; C\_\{\backslash alpha+1\}(x)\; =\; \backslash frac\{2\backslash alpha\}\{x\}\; C\_\backslash alpha(x)$

:$C\_\{\backslash alpha-1\}(x)\; +\; C\_\{\backslash alpha+1\}(x)\; =\; 2\backslash frac\{dC\_\backslash alpha\}\{dx\}$

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. In particular, it follows that:

:$\backslash int\_0^1\; x\; J\_\backslash alpha(x\; u\_\{\backslash alpha,m\})\; J\_\backslash alpha(x\; u\_\{\backslash alpha,n\})\; dx\; =\; \backslash frac\{\backslash delta\_\{m,n\}\}\{2\}\; [J\_\{\backslash alpha+1\}(u\_\{\backslash alpha,m\})]^2\; =\; \backslash frac\{\backslash delta\_\{m,n\}\}\{2\}\; [J\_\{\backslash alpha\}\text{'}(u\_\{\backslash alpha,m\})]^2,$

where α > −1, δ_m,n is the Kronecker delta, and u_α,m is the m-th zero of J_α(x). 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: :$\backslash int\_0^1\; x^2\; j\_\backslash alpha(x\; u\_\{\backslash alpha,m\})\; j\_\backslash alpha(x\; u\_\{\backslash alpha,n\})\; dx\; =\; \backslash frac\{\backslash delta\_\{m,n\}\}\{2\}\; [j\_\{\backslash alpha+1\}(u\_\{\backslash alpha,m\})]^2$

Another orthogonality relation is the closure equation:

:$\backslash int\_0^\backslash infty\; x\; J\_\backslash alpha(ux)\; J\_\backslash alpha(vx)\; dx\; =\; \backslash frac\{1\}\{u\}\; \backslash delta(u\; -\; v)$

for α > −1/2 and where δ is the Dirac delta function. This property is used to construct an arbitrary function from a series of Bessel functions by means of the Hankel transform. For the spherical Bessel functions the orthogonality relation is:

:$\backslash int\_0^\backslash infty\; x^2\; j\_\backslash alpha(ux)\; j\_\backslash alpha(vx)\; dx\; =\; \backslash frac\{\backslash pi\}\{2u^2\}\; \backslash delta(u\; -\; v)$

for α >  −1.

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

:$A\_\backslash alpha(x)\; \backslash frac\{dB\_\backslash alpha\}\{dx\}\; -\; \backslash frac\{dA\_\backslash alpha\}\{dx\}\; B\_\backslash alpha(x)\; =\; \backslash frac\{C\_\backslash alpha\}\{x\},$

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). 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

:$\backslash lambda^\{-\backslash nu\}\; J\_\backslash nu\; (\backslash lambda\; z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{n!\}\; \backslash left(\backslash frac\{(1-\backslash lambda^2)z\}\{2\}\backslash right)^n\; J\_\{\backslash nu+n\}(z)$

where $\backslash lambda$ and $\backslash nu$ may be taken as arbitrary complex numbers. A similar form may be given for $Y\_\backslash nu(z)$ and etc.

Bourget's hypothesis

Bessel himself originally proved that for non-negative integers n, the equation J_n(x) = 0 has an infinite number of solutions in x. When the functions J_n(x) are plotted on the same graph, though, none of the zeros seem to coincide for different values of n except for the zero at x = 0. This phenomenon is known as Bourget's hypothesis after the nineteenth century French mathematician who studied Bessel functions. Specifically it states that for any integers n ≥ 0 and m ≥ 1, the functions J_n(x) and J_n+m(x) have no common zeros other than the one at x = 0. The theorem was proved by Siegel in 1929.

Derivatives of J, Y, I, H, K

These formulas may be found in this reference.

p − 1 dependency

:$\backslash frac\{d\}\{dx\}y\_p(\backslash alpha\; x)=\backslash alpha\; y\_\{p-1\}(\backslash alpha\; x)\; -\; \backslash frac\{p\}\{x\}\; y\_p(\backslash alpha\; x)$ (note that the above equation is for $y\; =\; J,\; Y,\; I,\; H^\{(1)\},\; H^\{(2)\}$) :$\backslash frac\{d\}\{dx\}y\_p(\backslash alpha\; x)=-\backslash alpha\; y\_\{p-1\}(\backslash alpha\; x)\; -\; \backslash frac\{p\}\{x\}\; y\_p(\backslash alpha\; x)$ (note that the above equation is for $y\; =\; K$)

p + 1 dependency

:$\backslash frac\{d\}\{dx\}y\_p(\backslash alpha\; x)=-\backslash alpha\; y\_\{p+1\}(\backslash alpha\; x)\; +\; \backslash frac\{p\}\{x\}\; y\_p(\backslash alpha\; x)$ (note that the above equation is for $y\; =\; J,\; Y,\; K,\; H^\{(1)\},\; H^\{(2)\}$) :$\backslash frac\{d\}\{dx\}y\_p(\backslash alpha\; x)=\backslash alpha\; y\_\{p+1\}(\backslash alpha\; x)\; +\; \backslash frac\{p\}\{x\}\; y\_p(\backslash alpha\; x)$ (note that the above equation is for $y\; =\; I$)

Other relationships

:$\backslash frac\{d\}\{dx\}y\_p(\backslash alpha\; x)=\backslash frac\{\backslash alpha\}\{2\}[y\_\{p-1\}(\backslash alpha\; x)\; -\; y\_\{p+1\}(\backslash alpha\; x)]$ (note that the above equation is for $y\; =\; J,\; Y,\; H^\{(1)\},\; H^\{(2)\}$ only) :$y\_\{p-1\}(\backslash alpha\; x)\; +\; y\_\{p+1\}(\backslash alpha\; x)=\backslash frac\{2p\}\{\backslash alpha\; x\}y\_p(\backslash alpha\; x)$ (note that the above equation is for $y\; =\; J,\; Y,\; H^\{(1)\},\; H^\{(2)\}$ only)

Selected identities

$I\_\{-1/2\}\; \backslash left(z\backslash right)=\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\; z\}\}\backslash cosh(z)\; ;$

$I\_\{1/2\}\; \backslash left(z\backslash right)=\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\; z\}\}\backslash sinh(z)\; ;$

$I\_\backslash nu(z)=\backslash sum\_\{k=0\}\; \backslash frac\{z^k\}\{k!\}\; J\_\{\backslash nu+k\}(z);$

$J\_\backslash nu(z)=\backslash sum\_\{k=0\}\; (-1)^k\; \backslash frac\{z^k\}\{k!\}\; I\_\{\backslash nu+k\}(z);$

$I\_\backslash nu\; (\backslash lambda\; z)=\; \backslash lambda^\backslash nu\; \backslash sum\_\{k=0\}\; \backslash frac\{\backslash left(\backslash tfrac\{1\}\{2\}(\backslash lambda^2-1)z\backslash right)^k\}\{k!\}\; I\_\{\backslash nu+k\}(z);$

$I\_\backslash nu\; (z\_1+z\_2)=\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\; I\_\{\backslash nu-k\}(z\_1)I\_k(z\_2),\backslash quad\; J\_\backslash nu(z\_1\backslash pm\; z\_2)=\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\; J\_\{\backslash nu\; \backslash mp\; k\}(z\_1)J\_k(z\_2);$

$J\_\backslash nu(z)=\backslash frac\; z\; \{2\; \backslash nu\}\; (J\_\{\backslash nu-1\}(z)+J\_\{\backslash nu+1\}(z)),\; \backslash quad\; I\_\backslash nu(z)=\backslash frac\; z\; \{2\; \backslash nu\}\; (I\_\{\backslash nu-1\}(z)-I\_\{\backslash nu+1\}(z));$

$J\_\backslash nu\text{'}(z)=\backslash tfrac\{1\}\{2\}\; (J\_\{\backslash nu-1\}(z)-J\_\{\backslash nu+1\}(z)),\; \backslash quad\; I\_\backslash nu\text{'}(z)=\backslash tfrac\{1\}\{2\}(I\_\{\backslash nu-1\}(z)+I\_\{\backslash nu+1\}(z));$

$\backslash left(\backslash tfrac\{1\}\{2\}z\backslash right)^\backslash nu=\; \backslash Gamma(\backslash nu)\backslash cdot\; \backslash sum\_\{k=0\}\; I\_\{\backslash nu+2k\}(z)(\backslash nu+2k)\{-\backslash nu\backslash choose\; k\}$

= \Gamma(\nu)\cdot\sum_{k=0}(-1)^k J_{\nu+2k}(z)(\nu+2k){-\nu \choose k} = \Gamma(\nu+1)\cdot \sum_{k=0}\frac 1{k!}\left(\tfrac1 2z\right)^k J_{\nu+k}(z).

$K\_\backslash frac\{1\}\{2\}(z)=\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash mathrm\{e\}^\{-z\}z^\{-1/2\},\backslash ,\; z>0$

See also

Bessel–Clifford function

Bessel polynomials

Propagator

Fourier–Bessel series

Struve function

Kelvin functions

Lommel functions

Lommel polynomial

Neumann polynomial

Vibrations of a circular drum

Notes

References

Arfken, George B. and Hans J. Weber, Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005). ISBN 0-12-059876-0.

Bayin, S.S. Mathematical Methods in Science and Engineering, Wiley, 2006, Chapter 6.

Bayin, S.S., Essentials of Mathematical Methods in Science and Engineering, Wiley, 2008, Chapter 11.

Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4.

G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", Ann. Phys. Leipzig 25 (1908), p. 377.

B Spain, M.G. Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions.

Watson, G.N., A Treatise on the Theory of Bessel Functions, Second Edition, (1995) Cambridge University Press. ISBN 0-521-48391-3.

External links

Wolfram function pages on Bessel J and Y functions, and modified Bessel I and K functions. Pages include formulas, function evaluators, and plotting calculators.

Wolfram Mathworld – Bessel functions of the first kind

Category:Special functions Category:Special hypergeometric functions Category:Fourier analysis