The Bessel differential equation is a linear ordinary differential equation of the second order. It was named after the German mathematician Friedrich Wilhelm Bessel . Their solutions are called Bessel functions or cylinder functions .
Bessel differential equation
The Bessel differential equation is an ordinary linear differential equation of the second order, which is given by
is defined, where and are real or complex numbers . The solutions are called Bessel functions -th order.
Correspondingly, the Bessel operator is a second order differential operator . It is defined by
With it one can express Bessel's differential equation briefly by
Bessel functions
General
The solutions to Bessel's differential equation are called Bessel functions . They play an important role in physics, since the Bessel differential equation represents the radial part of the Laplace equation in the case of cylindrical symmetry . The Bessel functions are encountered, among other things, in the investigation of the natural vibrations of a circular membrane or an organ pipe, the propagation of water waves in round containers, the conduction of heat in rods, the analysis of the frequency spectrum of frequency- modulated signals, the field distribution in the cross-section of circular waveguides , the stationary ones States of box potentials, power distribution in nuclear reactors , the intensity of light diffraction at circular holes as well as filters in electrical engineering ( Bessel filters ). The Bessel functions are one of the special functions because of their diverse applications in mathematical physics .
As a differential equation of the second order of derivation, Bessel's differential equation has two linearly independent solutions . They can be described in different ways.
Bessel functions of the first kind
The first- order Bessel functions are defined as
-
,
where is the gamma function . In the origin ( ) these functions are finite for integers .
For non-integer are and linearly independent solutions.
The relationship applies to
integers
-
.
In this case, the second independent solution is the Bessel function of the second kind, which is discussed below.
Integral representations
For whole numbers, the Bessel function of the first kind can also be represented as an integral:
This is the -th Fourier coefficient of the function .
Hypergeometric function
The Bessel function of the first kind can be expressed by the generalized hypergeometric function :
This expression is related to the development of the Bessel function in relation to the Bessel-Clifford function .
Bessel functions of the second kind
The Bessel functions of the second kind (also called Weber functions or Neumann functions) also solve Bessel's differential equation. An alternative name is . For non-integers , they can be defined by
For integers is the function formed by the limit crossing
also a solution of Bessel's differential equation.
As for the Bessel functions of the first type, the following relationship also applies to the Bessel functions of the second type:
-
.
After crossing the border with the rule of de l'Hospital, this results
One finds explicit
for . Here is the Euler-Mascheroni constant and the -th harmonic number .
The Bessel functions of the second kind have a logarithmic singularity and a pole -th order.
For all of them , in addition to the Bessel function of the first type, the Bessel function of the second type is a second, linearly independent solution.
Bessel functions of the third kind
The Bessel functions of the third kind , (also known as Hankel are functions) linear combinations of Bessel functions of the first and second generic
where denotes the imaginary unit . These two functions are also linearly independent solutions of the Bessel differential equation.
properties
Relationships of orders of a genus
- For the Bessel functions , , and apply Rekursionsbeziehungen :
-
,
-
.
- For true .
- For true .
Asymptotic behavior
Let then hold for the asymptotic representations
For big arguments one finds
These formulas are for exact. Compare with the spherical Bessel functions below.
Modified Bessel functions
The differential equation
is solved by Bessel functions with a purely imaginary argument. The modified Bessel functions are normally defined for their solution
The function is also known as the MacDonald function . In contrast to the "normal" Bessel functions, the modified Bessel functions do not have an oscillating, but an exponential behavior.
Airy integrals
An integral representation can be given
for the functions and
-
.
Hypergeometric function
The modified Bessel function of the first kind can also be expressed by a generalized hypergeometric function :
-
.
Relationships of orders of a genus
- The recursion relationships apply to the Bessel functions and :
Asymptotic behavior
We again assume that is real and non-negative. For small arguments one finds
For big arguments you get
Spherical armchair functions
The Helmholtz equation in spherical coordinates leads to the radial equation after separating the variables
-
.
After the substitution
one obtains the Bessel differential equation
-
.
For the solution of the radial equation, the spherical Bessel functions , the spherical Neumann functions and the spherical Hankel functions are usually defined:
-
.
The alternative representations for
The spherical Bessel and Hankel functions are required, for example, for the treatment of the spherically symmetrical potential well in quantum mechanics .
properties
- For the spherical Bessel functions , , and apply Rekursionsbeziehungen :
-
.
-
.
Hankel transformation
The Hankel transform is an integral transform that is closely related to the Fourier transform . The integral kernel of the Hankel transformation is the Bessel function of the first kind , that is, the integral operator is:
-
.
A special property of the Hankel transformation is that it can be used to convert the Bessel operator into an algebraic expression (a multiplication).
history
Bessel functions were dealt with in detail by Bessel in 1824, but also appeared before that in the case of special physical problems, for example in Daniel Bernoulli (vibration of heavy chains 1738), Leonhard Euler (membrane vibration 1764), in celestial mechanics by Joseph-Louis Lagrange (1770 ) and with Pierre-Simon Laplace , in heat conduction with Joseph Fourier (heat propagation in the cylinder 1822) and Siméon Denis Poisson (1823).
literature
-
Milton Abramowitz , Irene Stegun : Handbook of Mathematical Functions. Dover, New York 1972, p. 355 .
- JH Graf, E. Gubler: Introduction to the theory of Bessel functions. First volume Second volume . KJ Wyss, Bern 1900
-
Carl Gottfried Neumann : Theory of Bessel's functions, an analogue to the theory of spherical functions. BG Teubner, Leipzig 1867.
-
Paul Schafheitlin The theory of Bessel functions . BG Teubner, Leipzig 1908.
-
GN Watson A Treatise on the Theory of Bessel functions , Cambridge University Press 1922, 1944, Archives
Bessel functions are dealt with in many textbooks in theoretical physics. B .:
-
John David Jackson : Classical Electrodynamics. John Wiley, New York NY 1962 (3rd edition. Ibid 1999, ISBN 0-471-30932-X ; German: 4th revised edition. De Gruyter, Berlin et al. 2006, ISBN 3-11-018970-4 ).
- Wolfgang Nolting: Basic course Theoretical Physics 5/2. Quantum Mechanics - Methods and Applications , 6th edition, Springer textbook, 2006, ISBN 978-3-540-26035-6
-
Arnold Sommerfeld Lectures on Theoretical Physics , Volume 6: Partial Differential Equations of Physics , Harri Deutsch 1992, ISBN 3-87144-379-4 .
Individual evidence
-
^ Bessel operator . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .
-
^ Friedrich Wilhelm Bessel: Investigation of the part of the planetary disturbances, which arises from the movement of the sun. In: Treatises of the Berlin Academy of Sciences 1824, Math. Classe, pp. 1–52, Berlin 1826.
-
^ Jacques Dutka: On the early history of Bessel functions. In: Archive for History of Exact Sciences . Volume 49, 1995, pp. 105-134.
-
^ GN Watson: Theory of Bessel Functions. Cambridge University Press, 1944, Chapter 1 (for history).
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