for an arbitrary real number α (the order). The most common and important special case is where α is an integer n.
Although α and α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g. so that the Bessel functions are analytic functions of α).

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = n; for spherical problems, one obtains half integer orders α = n+1/2.) For example:
Bessel functions also have useful properties for other problems, such as signal processing (e.g. see FM synthesis or Kaiser window).
Since 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, and the different variations are described below.
These are perhaps the most commonly used forms of the Bessel functions.
Y_{α}(x) is sometimes also called the Neumann function, and is occasionally denoted instead by N_{α}(x). It is related to J_{α}(x) by:
where the case of integer α is handled by taking the limit.
For integer order n, J_{n} and J_{n} are not linearly independent:
in which case Y_{n} is needed to provide the second linearly independent solution of Bessel's equation. In contrast, for noninteger order, J_{α} and J_{α} are linearly independent, and Y_{α} is redundant (as is clear from its definition above).
The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportional to 1/√x (see also their asymptotic forms, below), although their roots are not generally periodic except asymptotically for large x.
Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions H_{α}^{(1)}(x) and H_{α}^{(2)}(x), defined by:
where i is the imaginary unit. (The Hankel functions express inward and outwardpropagating cylindrical wave solutions of the cylindrical wave equation.)
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 of the first and second kind, and are defined by:
These are chosen to be realvalued for real arguments x. They are the two linearly independent solutions to the modified Bessel's equation:
Unlike the ordinary Bessel functions, which are oscillating, 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.
When solving for separable solutions of Laplace's equation in spherical coordinates, 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} (also denoted n_{n}), and are related to the ordinary Bessel functions J_{α} and Y_{α} by:
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_{n}^{(2)} is the complexconjugate of this (for real x). (!! is the double factorial.) It follows, for example, that j_{0}(x) = sin(x)/x and y_{0}(x) = cos(x)/x, and so on.
The Bessel functions have the following asymptotic forms. For small arguments 0 < x << 1, one obtains:
\frac{2}{\pi} \ln (x/2) & \mbox{if } \alpha=0 \\ \\ \frac{\Gamma(\alpha)}{\pi} \left( \frac{2}{x} \right) ^\alpha & \mbox{if } \alpha > 0\end{matrix} \right.</math>
where &alpha is nonnegative and Γ denotes the Gamma function. For large arguments x >> 1, they become:
\cos \left( x\frac{\alpha\pi}{2}  \frac{\pi}{4} \right)</math>
\sin \left( x\frac{\alpha\pi}{2}  \frac{\pi}{4} \right).</math>
Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations. For example, for large x >> 1, the modified Bessel functions become:
For integer order α = n, J_{n} is often defined via a Laurent series for a generating function:
an approach first 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 JacobiAnger identity:
which is used to expand a plane wave as a sum of cylindrical waves.
The functions J_{α}, Y_{α}, H_{α}^{(1)}, and H_{α}^{(2)} all satisfy the recurrence relations:
where Z denotes J, Y, H^{(1)}, or H^{(2)}. (These two identities are often combined to yield various other relations.)
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 FourierBessel 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.)
Another orthogonality relation is the closure equation:
for α > 1/2 and where δ is the Dirac delta function.
Another important consequence of the Hermitian nature of Bessel's equations 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). For example, if A_{α} = J_{α} and B_{α} = Y_{α}, then C_{α} is 2/&pi. This also holds for the modified Bessel functions; for example, if A_{α} = I_{α} and B_{α} = K_{α}, then C_{α} is 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.)
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