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 number α (the order). The most common and important special case is where α is an integer, n.

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

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Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = n; for 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.

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

Definitions

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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

Bessel functions of the first kind, denoted with J_α(x), are solutions of Bessel's differential equation which are finite at x = 0 for α an integer or α non-negative. The specific choice and normalization of J_α are defined by its properties below; another possibility is to define it by its Taylor series expansion around x = 0 (or a more general power series for non-integer α):

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

Here, $\backslash Gamma(z)$ is the gamma function, a generalization of the factorial 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.

If α is not an integer, the functions $J\_\backslash alpha\; (x)$ and $J\_\{-\backslash alpha\}\; (x)$ are linearly independent and are therefore the two solutions of the differential equation. On the other hand, if the order $\backslash alpha$ is an integer, then the following relationship is valid:

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

This means that they are no longer linearly independent. 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 $\backslash alpha$, is possible using an integral equation:

$J\_\backslash alpha\; (x)\; =\; \backslash frac\{1\}\{2\; \backslash pi\}\; \backslash int\_\{0\}^\{2\; \backslash pi\}\; \backslash cos\; (\backslash alpha\; \backslash tau\; -\; x\; \backslash sin\; \backslash tau)\; d\backslash tau.$

(For the full expression for real values of $\backslash alpha$, see Abramowitz and Stegun (1972) page 360)

This is the approach that Bessel used, and from this definition he derived several properties of the function. Another integral representation is:

$J\_\backslash alpha\; (x)\; =\; \backslash frac\{1\}\{2\; \backslash pi\}\; \backslash int\_\{-\backslash pi\}^\{\backslash pi\}\; e^\{i(\backslash alpha\; \backslash tau\; -\; x\; \backslash sin\; \backslash tau)\}\; d\backslash tau$

Relation to hypergeometric series

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

$J\_\backslash alpha(z)=\backslash frac\{(z/2)^\backslash alpha\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash ;\_0F\_1\; (\backslash alpha+1;\; -z^2/4).$

This expression is related to the development of Bessel functions in terms of the Bessel-Clifford function.

Bessel functions of the second kind

These are perhaps the most commonly used forms of the Bessel functions.

The Bessel functions of the second kind, denoted by Y_α(x), are solutions of the Bessel differential equation. They are singular (infinite) at x = 0.

Y_α(x) is sometimes also called the Neumann function, and is occasionally denoted instead by N_α(x). It is related to J_α(x) by:

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

where the case of integer α is handled by taking the limit.

When α is not an integer, the definition of Y_α is redundant (as is clear from its definition above). On the other hand, when α is an integer, Y_α 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:

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

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, 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.

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:

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

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

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. 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 for Hermann Hankel.

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:

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

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

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:

$I\_\backslash alpha(x)\; =\; i^\{-\backslash alpha\}\; J\_\backslash alpha(ix)\; \backslash !$

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

These are chosen to be real-valued for imaginary arguments x. They 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.

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

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

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

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

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

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

The spherical Bessel functions can also be written as:

$j\_n(x)\; =\; (-x)^n\; \backslash left(\backslash frac\{1\}\{x\}\backslash frac\{d\}\{dx\}\backslash right)^n\backslash ,\backslash frac\{\backslash sin\; x\}\{x\}\; ,$

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

The first spherical Bessel function $j\_0(x)$ is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:

$j\_0(x)=\backslash frac\{\backslash sin\; x\}\; \{x\}$

$j\_1(x)=\backslash frac\{\backslash sin\; x\}\; \{x^2\}-\; \backslash frac\{\backslash cos\; x\}\; \{x\}$

$j\_2(x)=\backslash left(\backslash frac\{3\}\; \{x^2\}\; -\; 1\; \backslash right)\backslash frac\{\backslash sin\; x\}\{x\}\; -\; \backslash frac\{3\backslash cos\; x\}\; \{x^2\}$

and

$y\_0(x)=-j\_\{-1\}(x)=-\backslash ,\backslash frac\{\backslash cos\; x\}\; \{x\}$

$y\_1(x)=j\_\{-2\}(x)=-\backslash ,\backslash frac\{\backslash cos\; x\}\; \{x^2\}-\; \backslash frac\{\backslash sin\; x\}\; \{x\}$

$y\_2(x)=-j\_\{-3\}(x)=\backslash left(-\backslash ,\backslash frac\{3\}\{x^2\}+1\; \backslash right)\backslash frac\{\backslash cos\; x\}\{x\}-\; \backslash frac\{3\; \backslash sin\; x\}\; \{x^2\}.$

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 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). (!! is the double factorial.) 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:

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

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

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

They satisfy the differential equation:

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

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

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

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

$Y\_\backslash alpha(x)\; \backslash rightarrow\; \backslash left\backslash \{\; \backslash begin\{matrix\}$

\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 (0.5772...) and Γ denotes the gamma function. For large arguments $x\; \backslash gg\; |\backslash alpha^2\; -\; 1/4|$, they become:

$J\_\backslash alpha(x)\; \backslash rightarrow\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\; x\}\}$

\cos \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right)

$Y\_\backslash alpha(x)\; \backslash rightarrow\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\; x\}\}$

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

$I\_\backslash alpha(x)\; \backslash rightarrow\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\; x\}\}\; e^x,$

$K\_\backslash alpha(x)\; \backslash rightarrow\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2x\}\}\; e^\{-x\}.$

while for small arguments , they become:

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

$K\_\backslash alpha(x)\; \backslash rightarrow\; \backslash left\backslash \{\; \backslash begin\{matrix\}$

- \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

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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 identity:

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

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.

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

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

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

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:

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

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

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:

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

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.)

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. 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 > 0.

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.)

See also

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• Bessel-Clifford function

References

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• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972)
  • Chapter 9 Bessel Functions of integer order
    • Section 9.1 J, Y (Weber) and H (Hankel)
    • Section 9.6 Modified (I and K)
    • Section 9.9 Kelvin functions
  • Chapter 10 Bessel Functions of fractional order
    • Section 10.1 Spherical Bessel Functions (j, y and h)
    • Section 10.2 Modified Spherical Bessel functions (I and K)
    • Section 10.3 Riccati-Bessel Functions
    • Section 10.4 Airy functions
• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).
• Frank Bowman, Introduction to Bessel Functions (Dover: New York, 1958) ISBN 0486604624.
• G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, (1966) Cambridge University Press.
• G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", Ann. Phys. Leipzig 25(1908), p.377.
• Hong Du, "Mie-scattering calculation," Applied Optics 43 (9), 1951-1956 (2004).

Special functions | Special hypergeometric functions

Besselsche Differentialgleichung | Fonction de Bessel | Funzioni di Bessel | ベッセル関数 | Funkcje Bessela | Функция Бесселя | Besslova funkcija | Besselfunktion | 贝塞尔函数