Bessel function

Bessel functions are therefore especially important for many problems of wave propagation and static potentials.

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.

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]

(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 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).

[1] Daniel considered a flexible chain suspended from a fixed point above and free at its lower end.

[1] Leonhard Euler in 1736, found a link between other functions (now known as Laguerre polynomials) and Bernoulli's solution.

[1] In the middle of the eighteen century, Jean le Rond d'Alembert had found a formula to solve the wave equation.

By 1771 there was dispute between Bernoulli, Euler, d'Alembert and Joseph-Louis Lagrange on the nature of the solutions vibrating strings.

, for integer n.[1] During the end of the 19th century Lagrange, Pierre-Simon Laplace and Marc-Antoine Parseval also found equivalents to the Bessel functions.

Friedrich Wilhelm Bessel had seen Lagrange solution but found it difficult to handle.

In 1813 in a letter to Carl Friedrich Gauss, Bessel simplified the calculation using trigonometric functions.