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**Bessel functions**, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are the canonical solutions *y*(*x*) of **Bessel's differential equation**

for an arbitrary complex number α (the **order** of the Bessel function). The most important cases are for α an integer or half-integer.

Although α and −α produce the same differential equation for real α, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α. Bessel functions are also known as **cylinder functions** or the **cylindrical harmonics** because they appear in the solution to Laplace's equation in cylindrical coordinates.

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*+1/2). 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
- Frequency-dependent friction in circular pipelines

Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).

Because 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. Different variations are described below.

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 or positive α, and diverge as *x* approaches zero for negative non-integer α. It is possible to define the function by its Taylor series expansion around *x* = 0:^{[1]}

where Γ(*z*) is the gamma function, a shifted generalization of the factorial function to non-integer values. The Bessel function of the first kind is an entire function if α is an integer. 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}*(

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 has simple poles at each of the non-positive integers):^{[2]}

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 definition may be extended to non-integer orders by (for Re(*x*) > 0)

^{[3]}^{[4]}^{[5]}^{[6]}

The Bessel functions can be expressed in terms of the generalized hypergeometric series as^{[7]}

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

The Bessel functions of the second kind, denoted by *Y*_{α}(*x*), occasionally denoted instead by *N*_{α}(*x*), are solutions of the Bessel differential equation that have a singularity at the origin (*x* = 0). These are sometimes called **Weber functions** after Heinrich Martin Weber, and also **Neumann functions** after Carl Neumann.

For non-integer α, *Y*_{α}(*x*) is related to *J*_{α}(*x*) by:

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

There is also a corresponding integral formula (for Re(*x*) > 0),^{[9]}

*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:

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.

Other choices are possible for the Bessel functions of the second kind, as any linearly independent solution may be used. This is the second alternative for the second solution in Fuchs's theorem when α is an integer.

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:^{[10]}

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:

If α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:^{[11]}

In particular, if α = *m* + 1/2 with *m* a nonnegative integer, the above relations imply directly that

These are useful in developing the spherical Bessel functions (below).

The Hankel functions admit the following integral representations for Re(*x*) > 0:^{[12]}

where the integration limits indicate integration along a contour that can be chosen as follows: from −∞ to 0 along the negative real axis, from 0 to ±iπ along the imaginary axis, and from ±iπ to +∞±iπ along a contour parallel to the real axis.^{[13]}

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:^{[14]}

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.

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if −π < arg(*z*) ≤ π/2):

*I*_{α}(*x*) and *K*_{α}(*x*) are the two linearly independent solutions to the **modified Bessel's equation**:^{[15]}

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.

Two integral formulas for the modified Bessel functions are (for Re(*x*) > 0):^{[16]}

Modified Bessel functions *K*_{1/3} and *K*_{2/3} can be represented in terms of rapidly converged integrals^{[17]}

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

**Basset function**after Alfred Barnard Basset**Modified Bessel function of the third kind****Modified Hankel function**^{[18]}**Macdonald function**after Hector Munro Macdonald**Weber function**^{[19]}**Neumann function**^{[19]}

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:^{[20]}

*y _{n}* is also denoted

The spherical Bessel functions can also be written as (**Rayleigh's Formulas**):^{[21]}

The first spherical Bessel function *j*_{0}(*x*) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:

^{[22]}

and

^{[23]}

The spherical Bessel functions have the generating functions ^{[24]}

In the following *f _{n}* is any of for

^{[25]}

There are also spherical analogues of the Hankel functions:

In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers *n*:

and is the complex-conjugate of this (for real *x*). It follows, for example, that and , and so on.

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:

They satisfy the differential equation:

This differential equation, and the Riccati–Bessel solutions, arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g., Du (2004)^{[26]} for recent developments and references.

Following Debye (1909), the notation is sometimes used instead of .

The Bessel functions have the following asymptotic forms for non-negative α. For small arguments , one obtains:^{[27]}

where γ is the Euler–Mascheroni constant (0.5772...) and Γ denotes the gamma function. For large real arguments , one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless α is half-integer) because they have zeros all the way out to infinity which would have to be matched exactly by any asymptotic expansion. However, for a given value of arg(*z*) one can write an equation containing a term of order |*z*|^{−1}:^{[28]}

(For α = 1/2 the last terms in these formulas drop out completely; see the spherical Bessel functions above.) 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 than by .

The asymptotic forms for the Hankel functions are:

These can be extended to other values of arg(*z*) using equations relating and to *H*_{α}^{(1)}(*z*) and *H*_{α}^{(2)}(*z*).^{[29]} It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, *J*_{α}(*z*) is not asymptotic to the average of these two asymptotic forms when *z* is negative (because one or the other will not be correct there, depending on the arg(*z*) used). 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)

For the modified Bessel functions, Hankel developed asymptotic expansions as well:

^{[30]}

^{[31]}

When α = 1/2 all the terms except the first vanish and we have

For small arguments , we have:

For integer order α = *n*, *J*_{n} is often defined via a Laurent series for a generating function:

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*:

and

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

is called Neumann expansion of *ƒ*. The coefficients for ν = 0 have the explicit form

where *O _{k}* is Neumann's polynomial.

Selected functions admit the special representation

with

due to the orthogonality relation

More generally, if *ƒ* has a branch-point near the origin of such a nature that

then

or

where is *f*'s Laplace transform.^{[33]}

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

where ν > −1/2 and *z* ∈ **C**.^{[34]} 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:

where *Z* denotes *J*, *Y*, *H*^{(1)}, or *H*^{(2)}. (These two identities are often combined, e.g. added or subtracted, to yield various other relations.) In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives). In particular, it follows that:

*Modified* Bessel functions follow similar relations :

and

The recurrence relation reads

where *C*_{α} denotes *I*_{α} or *e*^{απi}*K*_{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:

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*:^{[35]}

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:

for α > −1.

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

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.)

The Bessel functions obey a multiplication theorem

where λ and ν may be taken as arbitrary complex numbers. A similar form may be given for *Y*_{ν}(*z*) and so on.^{[36]}^{[37]}

Bessel himself originally proved that for non-negative integers *n*, the equation *J*_{n}(*x*) = 0 has an infinite number of solutions in *x*.^{[38]} 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}*(

- Anger function
- Bessel–Clifford function
- Bessel polynomials
- Fourier–Bessel series
- Frequency Modulation
- Hahn–Exton q-Bessel function
- Hankel transform
- Jackson q-Bessel function
- Kelvin functions
- Lerche–Newberger sum rule
- Lommel function
- Lommel polynomial
- Neumann polynomial
- Propagator
- Struve function
- Vibrations of a circular drum
- Weber function
- Wright generalized Bessel function

**^**Abramowitz and Stegun, p. 360, 9.1.10.**^**Abramowitz and Stegun, p. 358, 9.1.5.**^**Watson, p. 176**^**http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html**^**http://www.nbi.dk/~polesen/borel/node15.html**^**Arfken & Weber, exercise 11.1.17.**^**Abramowitz and Stegun, p. 362, 9.1.69.**^**Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.**^**Watson, p. 178.**^**Abramowitz and Stegun, p. 358, 9.1.3, 9.1.4.**^**Abramowitz and Stegun, p. 358, 9.1.6.**^**Abramowitz and Stegun, p. 360, 9.1.25.**^**Watson, p. 178**^**Abramowitz and Stegun, p. 375, 9.6.2, 9.6.10, 9.6.11.**^**Abramowitz and Stegun, p. 374, 9.6.1.**^**Watson, p. 181.**^**M.Kh.Khokonov.*Cascade Processes of Energy Loss by Emission of Hard Photons*, JETP, V.99, No.4, pp. 690-707 (2004). Derived from formulas sourced to I. S. Gradshteĭn and I. M. Ryzhik,*Table of Integrals, Series, and Products*(Fizmatgiz, Moscow, 1963; Academic, New York, 1980).**^**Referred to as such in: Teichroew, D.*The Mixture of Normal Distributions with Different Variances*, The Annals of Mathematical Statistics. Vol. 28, No. 2 (Jun., 1957), pp. 510–512- ^
^{a}^{b}http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf **^**Abramowitz and Stegun, p. 437, 10.1.1.**^**Abramowitz and Stegun, p. 439, 10.1.25, 10.1.26;**^**Abramowitz and Stegun, p. 438, 10.1.11.**^**Abramowitz and Stegun, p. 438, 10.1.12;**^**Abramowitz and Stegun, p. 439, 10.1.39.**^**Abramowitz and Stegun, p. 439, 10.1.23.**^**Hong Du, "Mie-scattering calculation,"*Applied Optics***43**(9), 1951–1956 (2004)**^**Arfken & Weber.**^**Abramowitz and Stegun, p. 364, 9.2.1;**^**NIST Digital Library of Mathematical Functions, Section 10.11.**^**Abramowitz and Stegun, p. 377, 9.7.1;**^**Abramowitz and Stegun, p. 378, 9.7.2;**^**Abramowitz and Stegun, p. 363, 9.1.82 ff.**^**E. T. Whittaker, G. N. Watson, A course in modern Analysis p. 536**^**I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors.*Table of Integrals, Series, and Products*, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Equation 8.411.10**^**Arfken & Weber, section 11.2**^**Abramowitz and Stegun, p. 363, 9.1.74.**^**C. Truesdell, "On the Addition and Multiplication Theorems for the Special Functions",*Proceedings of the National Academy of Sciences, Mathematics*, (1950) pp.752–757.**^**F. Bessel,*Untersuchung des Theils der planetarischen Störungen*, Berlin Abhandlungen (1824), article 14.**^**Watson, pp. 484–5**^**See, for example, Lide DR. CRC handbook of chemistry and physics: a ready-reference book of chemical CRC Press, 2004, ISBN 0-8493-0485-7, p. A-95

- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9",
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, New York: Dover, p. 355, ISBN 978-0486612720, MR 0167642 See also chapter 10. - 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. - Olver, F. W. J.; Maximon, L. C. (2010), "Bessel function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al.,
*NIST Handbook of Mathematical Functions*, Cambridge University Press, ISBN 978-0521192255, MR2723248 - Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.5. Bessel Functions of Integer Order",
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 - B Spain, M.G. Smith,
*Functions of mathematical physics*, Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions. - N. M. Temme,
*Special Functions. An Introduction to the Classical Functions of Mathematical Physics*, John Wiley and Sons, Inc., New York, 1996. ISBN 0-471-11313-1. 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.

- Lizorkin, P.I. (2001), "Bessel functions", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Karmazina, L.N.; Prudnikov, A.P. (2001), "Cylinder function", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Rozov, =N.Kh. (2001), "Bessel equation", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - 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
- Bessel functions J
_{ν}, Y_{ν}, I_{ν}and K_{ν}in Librow Function handbook.