Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then 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. Although α and −α produce the same differential equation, 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.
The most important cases are when α is an integer or halfinteger. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with halfinteger α are obtained when the Helmholtz equation is solved in spherical coordinates.
Applications of Bessel functions
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 halfinteger orders (α = n + 1/2). For example:
 Electromagnetic waves in a cylindrical waveguide
 Pressure amplitudes of inviscid rotational flows
 Heat conduction in a cylindrical object
 Modes of vibration of a thin circular (or annular) acoustic 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
 Frequencydependent friction in circular pipelines
 Dynamics of floating bodies
 Angular resolution
 Diffraction from helical objects, including DNA
Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).
Definitions
Because this is a secondorder differential equation, there must be two linearly independent solutions. 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.

Type First kind Second kind Bessel functions J_{α} Y_{α} Modified Bessel functions I_{α} K_{α} Hankel functions H^{(1)}
_{α} = J_{α} + iY_{α}H^{(2)}
_{α} = J_{α} − iY_{α}Spherical Bessel functions j_{n} y_{n} Spherical Hankel functions h^{(1)}
_{n} = j_{n} + iy_{n}h^{(2)}
_{n} = j_{n} − iy_{n}
Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by N_{n} and n_{n} respectively, rather than Y_{n} and y_{n}.^{[1]}^{[2]}
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 or positive α and diverge as x approaches zero for negative noninteger α. It is possible to define the function by its series expansion around x = 0, which can be found by applying the Frobenius method to Bessel's equation:^{[3]}
where Γ(z) is the gamma function, a shifted generalization of the factorial function to noninteger values. The Bessel function of the first kind is an entire function if α 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 functions that decay proportionally to (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large x. (The series indicates that −J_{1}(x) is the derivative of J_{0}(x), much like −sin x is the derivative of cos x; more generally, the derivative of J_{n}(x) can be expressed in terms of J_{n ± 1}(x) by the identities below.)
For noninteger 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 nonpositive integers):^{[4]}
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:^{[5]}
Another integral representation is:^{[5]}
This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to noninteger orders by one of Schläfli's integrals, for Re(x) > 0:^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}
The Bessel functions can be expressed in terms of the generalized hypergeometric series as^{[10]}
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 t, the Bessel function can be expressed as^{[11]}
Bessel functions of the second kind: Y_{α}
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) and are multivalued. These are sometimes called Weber functions, as they were introduced by H. M. Weber (1873), and also Neumann functions after Carl Neumann.^{[12]}
For noninteger 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 noninteger α tends to n:
If n is a nonnegative integer, we have the series^{[13]}
where is the digamma function, the logarithmic derivative of the gamma function.^{[14]}
There is also a corresponding integral formula (for Re(x) > 0):^{[15]}
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 at a nonzero value, then the Bessel functions are entire functions of α.
The Bessel functions of the second kind when α is an integer is an example of the second kind of solution in Fuchs's theorem.
Hankel functions: H^{(1)}
_{α}, H^{(2)}
_{α}
Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, H^{(1)}
_{α}(x) and H^{(2)}
_{α}(x), defined as^{[16]}
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 simplelooking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of the factor of the form e^{i f(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 inwardpropagating cylindricalwave 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:^{[17]}
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 (see below).
The Hankel functions admit the following integral representations for Re(x) > 0:^{[18]}
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.^{[15]}
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 as^{[19]}
when α is not an integer; when α is an integer, then the limit is used. These are chosen to be realvalued 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.
can be expressed in terms of Hankel functions:
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)^{[20]}:
I_{α}(x) and K_{α}(x) are the two linearly independent solutions to the modified Bessel's equation:^{[21]}
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 with the singularity being of logarithmic type.^{[22]}
Two integral formulas for the modified Bessel functions are (for Re(x) > 0):^{[23]}
In some calculations in physics, it can be useful to know that the following relation holds:
It can be proven by showing equality to the above integral definition for K_{0}. This is done by integrating a closed curve in the first quadrant of the complex plane.
Modified Bessel functions K_{1/3} and K_{2/3} can be represented in terms of rapidly convergent integrals^{[24]}
The modified Bessel function of the second kind has also been called by the following names (now rare):
 Basset function after Alfred Barnard Basset
 Modified Bessel function of the third kind
 Modified Hankel function^{[25]}
 Macdonald function after Hector Munro Macdonald
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
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^{[26]}
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)^{[27]}
The first spherical Bessel function j_{0}(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:^{[28]}
and^{[29]}
The spherical Bessel functions have the generating functions^{[30]}
In the following, f_{n} is any of j_{n}, y_{n}, h^{(1)}
_{n}, h^{(2)}
_{n} for n = 0, ±1, ±2, ...^{[31]}
Spherical Hankel functions: h^{(1)}
_{n}, h^{(2)}
_{n}
There are also spherical analogues of the Hankel functions:
In fact, there are simple closedform expressions for the Bessel functions of halfinteger order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for nonnegative integers n:
and h^{(2)}
_{n} is the complexconjugate of this (for real x). It follows, for example, that j_{0}(x) = sin x/x and y_{0}(x) = −cos x/x, 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: S_{n}, C_{n}, ξ_{n}, ζ_{n}
Riccati–Bessel functions only slightly differ from spherical Bessel functions:
They satisfy the differential equation
For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger's equation with hypothetical cylindrical infinite potential barrier.^{[32]} 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)^{[33]} for recent developments and references.
Following Debye (1909), the notation ψ_{n}, χ_{n} is sometimes used instead of S_{n}, C_{n}.
Asymptotic forms
The Bessel functions have the following asymptotic forms. For small arguments 0 < z ≪ √α + 1, one obtains, when α is not a negative integer:^{[3]}
When α is a negative integer, we have
For the Bessel function of the second kind we have three cases:
where γ is the Euler–Mascheroni constant (0.5772...).
For large real arguments z ≫ α^{2} − 1/4, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless α is halfinteger) 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}:^{[34]}
(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 H^{(1)}
_{α}(ze^{imπ}) and H^{(2)}
_{α}(ze^{imπ}) to H^{(1)}
_{α}(z) and H^{(2)}
_{α}(z).^{[35]}
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 (nonreal) z so long as z goes to infinity at a constant phase angle arg z (using the square root having positive real part):
For the modified Bessel functions, Hankel developed asymptotic (large argument) expansions as well:^{[36]}^{[37]}
When α = 1/2, all the terms except the first vanish, and we have
For small arguments 0 < z ≪ √α + 1, we have
Full domain approximations with elementary functions
Very good approximation (error below of the maximum value 1) of the Bessel function for an arbitrary value of the argument may be obtained with the elementary functions by joining the trigonometric approximation working for smaller values of with the expression containing attenuated cosine function valid for large arguments with a usage of the smooth transition function i.e.
Properties
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 noninteger 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 tonemodulated FM signal.
More generally, a series
is called Neumann expansion of f. The coefficients for ν = 0 have the explicit form
where O_{k} is Neumann's polynomial.^{[38]}
Selected functions admit the special representation
with
due to the orthogonality relation
More generally, if f has a branchpoint near the origin of such a nature that
then
or
where is the Laplace transform of f.^{[39]}
Another way to define the Bessel functions is the Poisson representation formula and the MehlerSonine formula:
where ν > −1/2 and z ∈ C.^{[40]} This formula is useful especially when working with Fourier transforms.
Because Bessel's equation becomes Hermitian (selfadjoint) 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 mth 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:
If one defines a boxcar function of x that depends on a small parameter ε as:
(where rect is the rectangle function) then the Hankel transform of it (of any given order α > −1/2), g_{ε}(k), approaches J_{α}(k) as ε approaches zero, for any given k. Conversely, the Hankel transform (of the same order) of g_{ε}(k) is f_{ε}(x):
which is zero everywhere except near 1. As ε approaches zero, the righthand side approaches δ(x − 1), where δ is the Dirac delta function. This admits the limit (in the distributional sense):
A change of variables then yields the closure equation:^{[41]}
for α > −1/2. The Hankel transform can express a fairly arbitrary function^{[clarification needed]}as an integral of Bessel functions of different scales. 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). In particular,
and
for α > −1.
For α > −1, the even entire function of genus 1, x^{−α}J_{α}(x), has only real zeros. Let
be all its positive zeros, then
(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 J_{α}, Y_{α}, H^{(1)}
_{α}, and H^{(2)}
_{α} all satisfy the recurrence relations^{[42]}
and
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^{[43]}
Modified Bessel functions follow similar relations:
and
The recurrence relation reads
where C_{α} denotes I_{α} or e^{αiπ}K_{α}. These recurrence relations are useful for discrete diffusion problems.
Multiplication theorem
The Bessel functions obey a multiplication theorem
where λ and ν may be taken as arbitrary complex numbers.^{[44]}^{[45]} For λ^{2} − 1 < 1,^{[44]} the above expression also holds if J is replaced by Y. The analogous identities for modified Bessel functions and λ^{2} − 1 < 1 are
and
Zeros of the Bessel function
Bourget's hypothesis
Bessel himself originally proved that for nonnegative integers n, the equation J_{n}(x) = 0 has an infinite number of solutions in x.^{[46]} 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 19thcentury 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 hypothesis was proved by Carl Ludwig Siegel in 1929.^{[47]}
Numerical approaches
For numerical studies about the zeros of the Bessel function, see Gil, Segura & Temme (2007), Kravanja et al. (1998) and Moler (2004).
See also
 Anger function
 Bessel–Clifford function
 Bessel–Maitland function
 Bessel polynomials
 Fourier–Bessel series
 Schlömilch's Series
 Hahn–Exton qBessel function
 Hankel transform
 Jackson qBessel function
 Kelvin functions
 KontorovichLebedev transform
 Lerche–Newberger sum rule
 Lommel function
 Lommel polynomial
 Neumann polynomial
 Sonine formula
 Struve function
 Vibrations of a circular drum
 Weber function
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