Bessel function

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 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.

times a Maclaurin series (note that α need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying the Frobenius method to Bessel's equation:[5]

Some earlier authors define the Bessel function of the first kind differently, essentially without the division by

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.

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:[8]

In terms of the Laguerre polynomials Lk and arbitrarily chosen parameter t, the Bessel function can be expressed as[16]

In the case of integer order n, the function is defined by taking the limit as a non-integer α tends to n:

Yα(x) is necessary as the second linearly independent solution of the Bessel's equation when α is an integer.

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.

These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations.

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).

Iα(x) and Kα(x) are the two linearly independent solutions to the modified Bessel's equation:[26]

Modified Bessel functions K1/3 and K2/3 can be represented in terms of rapidly convergent integrals[30]

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 jn and yn, and are related to the ordinary Bessel functions Jn and Yn by[32]

(For α = ⁠1/2⁠ the last terms in these formulas drop out completely; see the spherical Bessel functions above.)

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 integer order α = n, Jn is often defined via a Laurent series for a generating function:

arise in many physical systems and are defined in closed form by the Sung series.

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.

where α > −1, δm,n is the Kronecker delta, and uα,m is the mth zero of Jα(x).

(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):

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:

Bessel himself originally proved that for nonnegative integers n, the equation Jn(x) = 0 has an infinite number of solutions in x.

This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions.

[59] For numerical studies about the zeros of the Bessel function, see Gil, Segura & Temme (2007), Kravanja et al. (1998) and Moler (2004).