Bessel Function - Properties

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 non-integer 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 tone-modulated FM signal.

More generally, a series

is called Neumann expansion of ƒ. The coefficients for ν = 0 have the explicit form

where is Neumann's polynomial.

Selected functions admit the special representation

with

due to the orthogonality relation

More generally, if ƒ has a branch-point near the origin of such a nature that then

or

where is ƒ's Laplace transform.

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:

where ν > −1/2 and z is a complex number. This formula is useful especially when working with Fourier transforms.

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, 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 :

and

The recurrence relation reads

where C_α denotes I_α or eαπiK_α. 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, δ_m,n is the Kronecker delta, and u_{α, m} is the m-th 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:

Another orthogonality relation is the closure equation:

for α > −1/2 and where δ is the Dirac delta function. This property is used to construct an arbitrary function from a series of Bessel functions by means of the Hankel transform. 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). For example, if A_α = J_α and B_α = Y_α, then C_α is 2/π. 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.)