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Wikipedia / Bessel function

Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena with circular or cylindrical symmetry. They are named after the German astronomer and mathematician Friedrich Bessel, who studied them systematically in 1824.

Bessel functions are solutions to a particular type of ordinary differential equation is a number that determines the shape of the solution. This number is called the order, mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.

is an integer, the resulting Bessel functions are often called cylinder functions or cylindrical harmonics because they naturally arise when solving problems (like Laplace's equation) in cylindrical coordinates is a half-integer, the solutions are called spherical Bessel functions and are used in spherical systems, such as in solving the Helmholtz equation in spherical coordinates.

Applications

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). For example:

Bessel functions also appear in other fields, such as signal processing (e.g., see FM audio synthesis, Kaiser window, or Bessel filter). They also appear in pure mathematics as part of the Fourier expansion of Maass forms.

Definitions

Because this is a linear differential equation, solutions can be scaled to any amplitude. The amplitudes chosen for the functions originate from the early work in which the functions appeared as solutions to definite integrals rather than solutions to differential equations. Because the differential equation is second-order, there must be two linearly independent solutions: one of the first kind and one of the second kind. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections. The subscript n is known to be an integer.Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by and , respectively, rather than and .

Bessel functions of the first kind: Jα

Bessel functions of the first kind, denoted as , are solutions of Bessel's differential equation. For integer or positive, Bessel functions of the first kind are finite at the origin (); while for negative non-integer, Bessel functions of the first kind diverge as times a Maclaurin series (note that need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying the Frobenius method to Bessel's equation: where is the gamma function, a shifted generalization of the factorial; this definition is not used in this article. The Bessel function of the first kind is an entire function if is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large . (The series indicates that is the derivative of , much like is the derivative of ; more generally, the derivative of can be expressed in terms of by the identities below.)

For non-integer , the functions and are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order , the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):

This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

Bessel's integrals

Another definition of the Bessel function, for integer values of , is possible using an integral representation: which is also called Hansen-Bessel formula.

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 one of Schläfli's integrals, for :

Relation to hypergeometric series

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

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

Relation to Laguerre polynomials

In terms of the Laguerre polynomials and arbitrarily chosen parameter , the Bessel function can be expressed as

Bessel functions of the second kind: Yα

The Bessel functions of the second kind, denoted by , occasionally denoted instead by , are solutions of the Bessel differential equation that have a singularity at the origin () and are multivalued. These are sometimes called Weber functions, as they were introduced by , and also Neumann functions after Carl Neumann.

For non-integer , is related to

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

If is a nonnegative integer, we have the series is the digamma function, the logarithmic derivative of the gamma function.

There is also a corresponding integral formula (for ):

In the case where being Euler's constant

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

When

Both and are holomorphic functions of on the complex plane cut along the negative real axis. When is an integer, the Bessel functions are entire functions of . If is held fixed at a non-zero value, then the Bessel functions are entire functions of .

The Bessel functions of the second kind, when is an integer, are an example of the second kind of solution in Fuchs's theorem.

Hankel functions: Hα(1), Hα(2)

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, and , defined as where 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.

These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of Euler's formula, substituting , , as explicitly shown in the asymptotic expansion.

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).

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

In particular, if with

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

The Hankel functions admit the following integral representations for : 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 along the imaginary axis, and from to along a contour parallel to the real axis.

Modified Bessel functions: Iα, Kα

The Bessel functions are valid even for complex arguments , and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as when is not an integer. When is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments . The series expansion for is thus similar to that for , but without the alternating factor.

Using these two formulae the result to

given that the condition only when and but not when .

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):

and are the two linearly independent solutions to the modified Bessel's equation:

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, and are exponentially growing and decaying functions respectively. Like the ordinary Bessel function , the function goes to zero at for and is finite at for . Analogously, diverges at with the singularity being of logarithmic type for , and otherwise.

Two integral formulas for the modified Bessel functions are (for ):

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for

It can be proven by showing equality to the above integral definition for . This is done by integrating a closed curve in the first quadrant of the complex plane.

Modified Bessel functions of the second kind may be represented with Bassett's integral

Modified Bessel functions and can be represented in terms of rapidly convergent integrals

is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.

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

Spherical Bessel functions: jn, yn

When solving the Helmholtz equation in spherical coordinates by separation of variables

The two linearly independent solutions to this equation are called the spherical Bessel functions and , and are related to the ordinary Bessel functions and by

is also denoted or ; some authors call these functions the spherical Neumann functions.

The spherical Bessel functions can also be written as ()

The zeroth spherical Bessel function is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: and

The first few non-zero roots of the first few spherical Bessel functions are:

Generating function

The spherical Bessel functions have the generating functions

Finite series expansions

In contrast to the whole integer Bessel functions , the spherical Bessel functions have a finite series expression:

Differential relations

In the following, is any of , , , for

Spherical Hankel functions: hn(1), hn(2)

There are also spherical analogues of the Hankel functions

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 and is the complex-conjugate of this (for real ). 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: Sn, Cn, ξn, ζn

Riccati

For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger equation with hypothetical cylindrical infinite potential barrier. This differential equation, and the Riccati–Bessel solutions, also 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 , is sometimes used instead of , .

Asymptotic forms

The Bessel functions have the following asymptotic is not a negative integer:

When

where is the Euler–Mascheroni constant is imaginary.

For large real arguments , one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless 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 one can write an equation containing a term of order :

(For , the last terms in these formulas drop out completely; see the spherical Bessel functions above.)

These can be extended to other values of using equations relating and to and .

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, is not asymptotic to the average of these two asymptotic forms when is negative (because one or the other will not be correct there, depending on the 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) so long as goes to infinity at a constant phase angle (using the square root

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

)

When

Properties

is not a negative integer, the derivatives of the function can be defined as:

or, equivalently,

, a more general form of which is given below.

For integer order , 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.)

Infinite series arise in many physical systems and are defined in closed form by the Sung series.

A series expansion using Bessel functions (Kapteyn series

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.

is called Neumann expansion of . The coefficients for where is Neumann's polynomial.

More generally, if is the Laplace transform of .

where and . This formula is useful especially when working with Fourier transforms.

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by where , is the Kronecker delta, and is the th zero of . 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 for fixed and varying .

If one defines a boxcar function of that depends on a small parameter (where is the rectangle function) then the Hankel transform of it (of any given order ), , approaches as approaches zero, for any given . Conversely, the Hankel transform (of the same order) of is which is zero everywhere except near 1. As approaches zero, the right-hand side approaches , where is the Dirac delta function. This admits the limit (in the distributional

A change of variables then yields the closure equation: for for .

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian where and are any two solutions of Bessel's equation, and is a constant independent of for .

For , the even entire function of genus 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.)

Recurrence relations

The functions , , , and all satisfy the recurrence relations where denotes , , , or . 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

Using the previous relations one can arrive to similar relations for the Spherical Bessel functions:

and

Modified

where denotes or . These recurrence relations are useful for discrete diffusion problems.

Transcendence

In 1929, Carl Ludwig Siegel proved that , , and the logarithmic derivative are transcendental numbers when ν is rational and x is algebraic and nonzero. is transcendental under the same assumptions.

Sums with Bessel functions

Multiplication theorem

The Bessel functions obey a multiplication theorem where and may be taken as arbitrary complex numbers. For , the above expression also holds if is replaced by . The analogous identities for modified Bessel functions and

Zeros of the Bessel function

Bourget's hypothesis

Bessel himself originally proved that for nonnegative integers , the equation has an infinite number of solutions in . When the functions are plotted on the same graph, though, none of the zeros seem to coincide for different values of except for the zero at . This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers and , the functions and have no common zeros other than the one at . The hypothesis was proved by Carl Ludwig Siegel in 1929.

Transcendence

Siegel proved in 1929 that when ν is rational, all nonzero roots of and are transcendental, as are all the roots of . for .

Numerical approaches

For numerical studies about the zeros of the Bessel function, see , and .

Numerical values

The first zeros in J0 (i.e., j0,1, j0,2 and j0,3) occur at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.

History

Waves and elasticity problems

The first appearance of a Bessel function appears in the work of Daniel Bernoulli in 1732, while working on the analysis of a vibrating string, a problem that was tackled before by his father Johann Bernoulli. Daniel considered a flexible chain suspended from a fixed point above and free at its lower end.. Bernoulli also developed a method to find the zeros of the function.

Leonhard Euler in 1736, found a link between other functions (now known as Laguerre polynomials.

In the middle of the eighteen century, Jean le Rond d'Alembert had found a formula to solve the wave equation. By 1771 there was dispute between Bernoulli, Euler, d'Alembert and Joseph-Louis Lagrange on the nature of the solutions of vibrating strings.

Euler worked in 1778 on buckling, introducing the concept of Euler's critical load., for integer n.

During the end of the 18th century Lagrange, Pierre-Simon Laplace and Marc-Antoine Parseval also found equivalents to the Bessel functions. using cosine.

At the beginning of the 1800s, Joseph Fourier to solve the heat equation in a problem with cylindrical symmetry. Fourier won a prize of the French Academy of Sciences for this work in 1811. But most of the details of his work, including the use of a Fourier series, remained unpublished until 1822. Poisson, in rivalry with Fourier, extended Fourier's work in 1823, introducing new properties of Bessel functions, including Bessel functions of half-integer order (now known as spherical Bessel functions).

Astronomical problems

In 1770, Lagrange introduced the series expansion of Bessel functions to solve Kepler's equation, a transcendental equation in astronomy. Friedrich Wilhelm Bessel had seen Lagrange's solution but found it difficult to handle. In 1813 in a letter to Carl Friedrich Gauss, Bessel simplified the calculation using trigonometric functions. Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier which was published later. In 1824, Bessel carried out a systematic investigation of the functions, which earned the functions his name. In older literature the functions were called cylindrical functions or even Bessel–Fourier functions.

See also

Notes

References

  • Arfken, George B. and Hans J. Weber, Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005). .

  • Reproduced as pages 84 to 109 in English translation of the text.

  • Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). .

  • .

  • .

  • 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. . Chapter 9 deals with Bessel functions.

  • Watson, G. N., A Treatise on the Theory of Bessel Functions, Second Edition, (1995) Cambridge University Press. .

  • .

  • .

  • .

  • .

  • 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 page on Riccati-Bessel functions

  • Bessel functions J, Y, I and K in Librow Function handbook.

  • F. W. J. Olver, L. C. Maximon, Bessel Functions (chapter 10 of the Digital Library of Mathematical Functions).