Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation
The most important cases are when is an integer or half-integer. Bessel functions for integer 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 half-integer are obtained when the Helmholtz equation is solved in spherical coordinates.
The Bessel function is a generalization of the sine function. It can be interpreted as the vibration of a string with variable thickness, variable tension (or both conditions simultaneously); vibrations in a medium with variable properties; vibrations of the disc membrane, etc.
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:
Bessel functions also appear in other problems, such as signal processing (e.g., see FM audio synthesis, Kaiser window, or Bessel filter).
Because this is a second-order linear 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, denoted as J_{α}(x), are solutions of Bessel's differential equation. For integer or positive α, Bessel functions of the first kind are finite at the origin (x = 0); while for negative non-integer α, Bessel functions of the first kind diverge as x approaches zero. 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]}
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 n, the following relationship is valid (the gamma function has simple poles at each of the non-positive 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]}
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 Re(x) > 0:^{[5]}^{[7]}^{[8]}^{[9]}^{[10]}
The Bessel functions can be expressed in terms of the generalized hypergeometric series as^{[11]}
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^{[12]}
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.^{[13]}
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:
If n is a nonnegative integer, we have the series^{[14]}
where is the digamma function, the logarithmic derivative of the gamma function.^{[15]}
There is also a corresponding integral formula (for Re(x) > 0):^{[16]}
In the case where n = 0,
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 non-zero 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.
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^{[17]}
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.
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 e^{i f(x)}. For real where , 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 H^{(1)}
_{α}(x), H^{(2)}
_{α}(x) for and , for , , 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).
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:^{[18]}
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:^{[19]}
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^{[20]}
can be expressed in terms of Hankel functions:
Using these two formulae the result to +, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following
given that the condition Re(x) > 0 is met. It can also be shown that
only when |Re(α)| < 1/2 and Re(x) ≥ 0 but not when x = 0.^{[21]}
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):^{[22]}
I_{α}(x) and K_{α}(x) are the two linearly independent solutions to the modified Bessel's equation:^{[23]}
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 for K_{0}, and 1/2Γ(|α|)(2/x)^{|α|} otherwise.^{[24]}
Two integral formulas for the modified Bessel functions are (for Re(x) > 0):^{[25]}
Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example:
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^{[26]}
The modified Bessel function 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):
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^{[28]}
y_{n} is also denoted n_{n} or η_{n}; some authors call these functions the spherical Neumann functions.
From the relations to the ordinary Bessel functions it is directly seen that:
The spherical Bessel functions can also be written as (Rayleigh's formulas)^{[29]}
The zeroth spherical Bessel function j_{0}(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:^{[30]}
The spherical Bessel functions have the generating functions^{[32]}
In the following, f_{n} is any of j_{n}, y_{n}, h^{(1)}
_{n}, h^{(2)}
_{n} for n = 0, ±1, ±2, …^{[33]}
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 h^{(2)}
_{n} is the complex-conjugate 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 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.^{[34]} 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)^{[35]} for recent developments and references.
Following Debye (1909), the notation ψ_{n}, χ_{n} is sometimes used instead of S_{n}, C_{n}.
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:
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 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}:^{[36]}
(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
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).^{[37]}
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) 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:^{[38]}^{[39]}
There is also the asymptotic form (for large real )^{[40]}
When α = 1/2, all the terms except the first vanish, and we have
For small arguments 0 < |z| ≪ √α + 1, we have
For integer order α = n, J_{n} is often defined via a Laurent series for a generating function:
A series expansion using Bessel functions (Kapteyn series) is
Another important relation for integer orders is the Jacobi–Anger expansion:
More generally, a series
Selected functions admit the special representation
More generally, if f has a branch-point near the origin of such a nature that
Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:
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:
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:
A change of variables then yields the closure equation:^{[44]}
Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:
For α > −1, the even entire function of genus 1, x^{−α}J_{α}(x), has only real zeros. Let
(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 functions J_{α}, Y_{α}, H^{(1)}
_{α}, and H^{(2)}
_{α} all satisfy the recurrence relations^{[45]}
Modified Bessel functions follow similar relations:
The recurrence relation reads
In 1929, Carl Ludwig Siegel proved that J_{ν}(x), J'_{ν}(x), and the quotient J'_{ν}(x)/J_{ν}(x) are transcendental numbers when ν is rational and x is algebraic and nonzero.^{[47]} The same proof also implies that K_{ν}(x) is transcendental under the same assumptions.^{[48]}
The Bessel functions obey a multiplication theorem
Bessel himself originally proved that for nonnegative integers n, the equation J_{n}(x) = 0 has an infinite number of solutions in x.^{[51]} 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 19th-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 hypothesis was proved by Carl Ludwig Siegel in 1929.^{[52]}
Siegel proved in 1929 that when ν is rational, all nonzero roots of J_{ν}(x) and J'_{ν}(x) are transcendental,^{[53]} as are all the roots of K_{ν}(x).^{[48]} It is also known that all roots of the higher derivatives for n ≤ 18 are transcendental, except for the special values and .^{[53]}
For numerical studies about the zeros of the Bessel function, see Gil, Segura & Temme (2007), Kravanja et al. (1998) and Moler (2004).
The first zero in J_{0} (i.e, j_{0,1}, j_{0,2} and j_{0,3}) occurs at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.^{[54]}