In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y ( x ) of Bessel's differential equation:

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 mostly smooth functions of α ).

Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical 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

Bessel functions also have useful properties for other problems, such as signal processing ( e. g., see FM synthesis, Kaiser window, or Bessel filter ).

Bessel functions of the first kind, denoted as J < sub > α </ sub >( x ), are solutions of Bessel's differential equation that are finite at the origin ( x = 0 ) for integer α, and diverge as x approaches zero for negative non-integer α.

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.

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 Y < sub > α </ sub >( x ), occasionally denoted instead by N < sub > α </ sub >( x ), are solutions of the Bessel differential equation.

When α is an integer, the Bessel functions J are entire functions of x.

If x is held fixed, then the Bessel functions are entire functions of α.

In terms of the Laguerre polynomials and arbitrarily chosen parameter, the Bessel function can be expressed as

The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.

) 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 ).

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 < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

The harmonic distribution of a sine wave carrier modulated by such a sinusoidal signal can be represented with Bessel functions ; this provides the basis for a mathematical understanding of frequency modulation in the frequency domain.

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number and the modulation index.

For more general distributions the asymptotic distribution can be expressed as a Bessel function.

Second-order filters can have a Bessel, Linkwitz-Riley or Butterworth characteristic depending on design choices and the components used.

Formula for spectral distribution of synchrotron radiation, given above, can be expressed in terms of rapidly coverged integral with no special functions involved ( see also modified Bessel functions ) by means of the relation:

The other difference formulas, such as those of Stirling, Bessel and Gauss, can be derived from Newton's, using Newton's terms, with data points and x values renamed in keeping with the choice of x zero, and based on the fact that they must add up to the same sum value as Newton's ( With Stirling that is so when polynomial degree is even.

In 1980 he and Nobel laureate Edwin McMillan found a new algorithm for high-precision computation of the Euler-Mascheroni constant using Bessel functions, and showed that can not have a simple rational form p / q ( where p and q are integers ) unless q is extremely large ( greater than 10 < sup > 15000 </ sup >).

Note also that, using the limiting values of the modified Bessel function for small arguments, we can recover the Poisson distribution as a special case of the Skellam distribution for.

The probability density can be expressed as a series of Bessel functions ( see Abramowitz and Stegun § 9. 6. 34 )

For example, the solution for the time-dependent " start-up " process can be expressed in terms of Bessel functions.

so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions.

This is the greatest number of terms that can be set to zero, since there are a total of four coefficients in the third order Bessel polynomial, requiring four equations in order to be defined.

Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:

In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

Bessel roughly estimated the period of the companion of Sirius to be about half a century ; C. A. F. Peters computed an orbit for it in 1851.

Streams of vapour observed during the comet's 1835 apparition prompted astronomer Friedrich Wilhelm Bessel to propose that the jet forces of evaporating material could be great enough to significantly alter a comet's orbit.

Examples would be crossover slope, filter type ( e. g., Bessel, Butterworth, etc.

In regards to the stars, Tycho also believed that if the Earth orbited the Sun annually there should be an observable stellar parallax over any period of six months, during which the angular orientation of a given star would change thanks to Earth's changing position ( this parallax does exist, but is so small it was not detected until 1838, when Friedrich Bessel discovered a parallax of 0. 314 arcseconds of the star 61 Cygni in 1838 ).

The physical argument that ψ must be defined everywhere selected Bessel function of the first kind J over the other possibilities in the vacuum case.

The Board of Trustees resolved that it would be named for his father, Franz Theodore Stone, a Prussian mathematician and astronomical researcher who worked for Friedrich Wilhelm Bessel after attending the Königsberg Albertina University.

This is a general property of the group delay for a Bessel filter of order n: the first terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at.

Crick, however, knowing the Fourier transforms of Bessel functions that represent the X-ray diffraction patterns of helical structures of atoms, correctly interpreted further one of Dr. Franklin's experimental findings as indicating that DNA was most likely to be a double helix with the two polynucleotide chains running in opposite directions.

The terms in this expansion are spherical harmonics ( which give the angular dependence ) multiplied by spherical Bessel functions ( which give the radial dependence ).

* Wright generalized Bessel function

Generalized hypergeometric functions include the ( Gaussian ) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.

The functions of the form are called confluent hypergeometric limit functions and are closely related to Bessel functions.