for an arbitrary complex number, which represents the order of the Bessel function. Although and 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 .
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:
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.
Bessel functions of the first kind, denoted as Jα(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:
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, 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 −J1(x) is the derivative of J0(x), much like −sin x is the derivative of cos x; more generally, the derivative of Jn(x) can be expressed in terms of Jn ± 1(x) by the identities below.)
For non-integer α, the functions Jα(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):
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.
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:
The Bessel functions of the second kind, denoted by Yα(x), occasionally denoted instead by Nα(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.
For non-integer α, Yα(x) is related to Jα(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
There is also a corresponding integral formula (for Re(x) > 0):
In the case where n = 0,
Yα(x) is necessary as the second linearly independent solution of the Bessel's equation when α is an integer. But Yα(x) has more meaning than that. It can be considered as a "natural" partner of Jα(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 Jα(x) and Yα(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
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 eif(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:
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:
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 ±πi along the imaginary axis, and from ±πi to +∞ ± πi along a contour parallel to the real axis.
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
when α is not an integer; when α is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments x. The series expansion for Iα(x) is thus similar to that for Jα(x), but without the alternating (−1)m factor.
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.
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):
Iα(x) and Kα(x) 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, Iα and Kα are exponentially growing and decaying functions respectively. Like the ordinary Bessel function Jα, the function Iα goes to zero at x = 0 for α > 0 and is finite at x = 0 for α = 0. Analogously, Kα diverges at x = 0 with the singularity being of logarithmic type for K0, and 1/2Γ(|α|)(2/x)|α| otherwise.
Two integral formulas for the modified Bessel functions are (for Re(x) > 0):
Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for Re(ω) > 0):
It can be proven by showing equality to the above integral definition for K0. This is done by integrating a closed curve in the first quadrant of the complex plane.
Modified Bessel functions K1/3 and K2/3 can be represented in terms of rapidly convergent integrals
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):
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 j0(x) = sin x/x and y0(x) = −cos x/x, and so on.
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. 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 ψn, χn is sometimes used instead of Sn, Cn.
For large real arguments z ≫ |α2 − 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:
(For α = 1/2 the last terms in these formulas drop out completely; see the spherical Bessel functions above.)
The asymptotic forms for the Hankel functions are:
These can be extended to other values of arg z using equations relating H(1) α(zeimπ) and H(2) α(zeimπ) to H(1) α(z) and H(2) α(z).
It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, Jα(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):
Infinite series of Bessel functions in the form where arise in many physical systems and defined in closed form by the Sung series. For example, when N = 3: . More generally, the Sung series and the alternating Sung series are written as:
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, δm,n is the Kronecker delta, and uα,m is the mth zero of Jα(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 Jα(xuα,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), gε(k), approaches Jα(k) as ε approaches zero, for any given k. Conversely, the Hankel transform (of the same order) of gε(k) is fε(x):
which is zero everywhere except near 1. As ε approaches zero, the right-hand 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:
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 Aα and Bα are any two solutions of Bessel's equation, and Cα is a constant independent of x (which depends on α and on the particular Bessel functions considered). In particular,
for α > −1.
For α > −1, the even entire function of genus 1, x−αJα(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.)
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:
The recurrence relation reads
where Cα denotes Iα or eαiπKα. These recurrence relations are useful for discrete diffusion problems.
In 1929, Carl Ludwig Siegel proved that Jν(x), J'ν(x), and the quotient J'ν(x)/Jν(x) are transcendental numbers when ν is rational and x is algebraic and nonzero. The same proof also implies that Kν(x) is transcendental under the same assumptions.
where λ and ν may be taken as arbitrary complex numbers. For |λ2 − 1| < 1, the above expression also holds if J is replaced by Y. The analogous identities for modified Bessel functions and |λ2 − 1| < 1 are
Bessel himself originally proved that for nonnegative integers n, the equation Jn(x) = 0 has an infinite number of solutions in x. When the functions Jn(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 Jn(x) and Jn + m(x) have no common zeros other than the one at x = 0. The hypothesis was proved by Carl Ludwig Siegel in 1929.
Siegel proved in 1929 that when ν is rational, all nonzero roots of Jν(x) and J'ν(x) are transcendental, as are all the roots of Kν(x). It is also known that all roots of the higher derivatives for n ≤ 18 are transcendental, except for the special values and .
^Wilensky, Michael; Brown, Jordan; Hazelton, Bryna (June 2023). "Why and when to expect Gaussian error distributions in epoch of reionization 21-cm power spectrum measurements". Monthly Notices of the Royal Astronomical Society. 521 (4): 5191–5206. arXiv:2211.13576. doi:10.1093/mnras/stad863.