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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:
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 Jα(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:
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 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 −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 α, the following relationship is valid (note that 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.
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), one of Schäfli's integrals:
This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.
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). Bessel functions of the second kind are multivalued with singularity at zero. These are sometimes called Weber functions after Heinrich Martin Weber, 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:
There is also a corresponding integral formula (for Re(x) > 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 Hα(1)(x) and Hα(2)(x), defined by:
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 eif(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:
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:
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.
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:
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.
If −π < arg(x) ≤ π/2, Kα(x) can be expressed as a Hankel function of the first kind:
and if π/2 < arg(x) ≤ π, it can be expressed as a Hankel function of the second kind:
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.
Two integral formulas for the modified Bessel functions are (for Re(x) > 0):
Modified Bessel functions K1/3 and K2/3 can be represented in terms of rapidly converged integrals
The modified Bessel function of the second kind has also been called by the now-rare names:
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 jn and yn, and are related to the ordinary Bessel functions Jn and Yn by:
yn is also denoted nn or ηn; some authors call these functions the spherical Neumann functions.
The spherical Bessel functions can also be written as (Rayleigh's formulas):
The first spherical Bessel function j0(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:
The spherical Bessel functions have the generating functions 
In the following fn is any of for 
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) for recent developments and references.
Following Debye (1909), the notation is sometimes used instead of .
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 , 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.) Even though these equations are true, better approximations may be available for complex z. For example, J0(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 and to Hα(1)(z) and Hα(2)(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):
When α = 1/2 all the terms except the first vanish and we have
For small arguments , we have:
For integer order α = n, Jn 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:
More generally, a series
is called Neumann expansion of ƒ. The coefficients for ν = 0 have the explicit form
Selected functions admit the special representation
due to the orthogonality relation
More generally, if ƒ 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:
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 eαπiKα. 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, δm,n is the Kronecker delta, and uα, m is the m-th 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(x uα, 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 greater than −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. So by abuse of language (or "formally"), one says that
even though the integral on the left is not actually defined. A change of variables then yields the closure equation:
for α > −1/2. The Hankel transform can express a fairly arbitrary function as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:
for α > −1. Again, this is a useful formal equation whose left-hand side is not actually defined.
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,
(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
Bessel himself originally proved that for non-negative 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 nineteenth 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.