Bessel functions and diffraction by helical structures
The linear differential equation
 |
(0.1) |
in which
is a constant (non-negative if it is real), is called the Bessel's equation. We derive its general solution by trying the series form
 |
(0.2) |
due to Frobenius. Since the parameter
is indefinite, we may regard
as distinct from 0.
We substitute (2) and the derivatives of the series in (1):
Thus the coefficients of the powers
,
,
and so on must vanish, and we get the system
of equations
![\begin{align*}\begin{cases}{[}r^2-p^2{]}a_0 = 0,\\ {[}(r+1)^2-p^2{]}a_1 = 0,\\ {...
...\qquad \qquad \ldots\\ {[}(r+k)^2-p^2{]}a_k+a_{k-2} = 0. \end{cases}\end{align*}](img10.png) |
(0.3) |
The last of those can be written
Because
, the first of those (the indicial equation) gives
, i.e. we have the roots
Let's first look the the solution of (1) with
; then
, and thus
From the system (3) we can solve one by one each of the coefficients
,
,
and express them with
which remains arbitrary. Setting for
the integer values we get
 |
(0.4) |
(where
).
Putting the obtained coefficients to (2) we get the particular solution
![$\displaystyle y_1 := a_0x^p \left[1\!-\!\frac{x^2}{2(2p\!+\!2)}\! +\!\frac{x^4}...
...frac{x^6}{2\!\cdot\!4\!\cdot\!6(2p\!+\!2)(2p\!+\!4)(2p\!+\!6)}\!+-\ldots\right]$](img25.png) |
(0.5) |
In order to get the coefficients
for the second root
we have to look after that
or
. Therefore
where
is a positive integer. Thus, when
is not an integer and not an integer added by
, we get the second particular solution, gotten of (5) by replacing
by
:
![$\displaystyle y_2 := a_0x^{-p}\!\left[1 \!-\!\frac{x^2}{2(-2p\!+\!2)}\!+\!\frac...
...c{x^6}{2\!\cdot\!4\!\cdot\!6(-2p\!+\!2)(-2p\!+\!4)(-2p\!+\!6)}\!+-\ldots\right]$](img36.png) |
(0.6) |
The power series of (5) and (6) converge for all values of
and are linearly independent (the ratio
tends to 0 as
). With the appointed value
the solution
is called the Bessel function of the first kind and of order
and denoted by
. The similar definition is set for the first kind Bessel function of an arbitrary order
(and
).
For
the general solution of the Bessel's differential equation is thus
where
with
.
The explicit expressions for
are
 |
(0.7) |
which are obtained from (5) and (6) by using the last formula for gamma function.
E.g. when
the series in (5) gets the form
Thus we get
analogically (6) yields
and the general solution of the equation (1) for
is
In the case that
is a non-negative integer
, the “+” case of (7) gives the solution
but for
the expression of
is
, i.e. linearly dependent of
. It can be shown that the other solution of (1) ought to be searched in the form
. Then the general solution is
.
Other formulae
The first kind Bessel functions of integer order have the generating function
:
 |
(0.8) |
This function has an essential singularity at
but is analytic elsewhere in
; thus
has the Laurent expansion in that point. Let us prove (8) by using the general expression
of the coefficients of Laurent series. Setting to this
,
,
gives
The paths
and
go once round the origin anticlockwise in the
-plane and
-plane, respectively. Since the residue of
in the origin is
, the residue theorem gives
This means that
has the Laurent expansion (8).
By using the generating function, one can easily derive other formulae, e.g.
the integral representation of the Bessel functions of integer order:
Also one can obtain the addition formula
and the series representations of cosine and sine:
One notes also that Bessel's equation arises in the derivation of separable solutions to Laplace's equation, and also for the Helmholtz equation in either cylindrical or spherical coordinates. The Bessel functions are therefore very important in many physical problems involving wave propagation, wave diffraction phenomena-including X-ray
diffraction by certain molecular crystals, and also static potentials. The solutions to most problems in cylindrical coordinate systems are found in terms of Bessel functions of integer order (
), whereas in spherical coordinates, such solutions involve Bessel functions of half-integer orders (
).
Several examples of Bessel function solutions are:
- the diffraction pattern of a helical molecule
wrapped around a cylinder computed from the Fourier transform
of the helix in cylindrical coordinates;
- electromagnetic waves in a cylindrical waveguide
- diffusion problems on a lattice.
- vibration modes of a thin circular, tubular or annular membrane (such as a drum, other membranophone, the vocal cords, etc.)
- heat
conduction
in a cylindrical object
In engineering Bessel functions also have useful properties for signal processing and filtering noise as for example by using Bessel filters, or in FM synthesis and windowing signals.
The first example listed above was shown to be especially important in molecular
biology for the structures of helical secondary structures in certain proteins (e.g.
) or in molecular genetics for finding the double-helix
structure of Deoxyribonucleic Acid (DNA) molecular crystals with extremely important consequences for genetics, biology, mutagenesis, molecular evolution,
contemporary life sciences and medicine. This finding is further detailed in a related entry.
- 1
-
F. Bessel, ``Untersuchung des Theils der planetarischen Störungen'', Berlin Abhandlungen (1824), article 14.
- 2
-
Franklin, R.E. and Gosling, R.G. received. 6th March 1953. Acta Cryst. (1953). 6, 673 The Structure of Sodium Thymonucleate Fibres I. The Influence of Water Content Acta Cryst. (1953). 6,678 : The Structure of Sodium Thymonucleate Fibres II. The Cylindrically Symmetrical Patterson Function.
- 3
-
Arfken, George B. and Hans J. Weber, Mathematical Methods for Physicists, 6th edition, Harcourt: San Diego, 2005. ISBN 0-12-059876-0.
- 4
-
Bowman, Frank. Introduction to Bessel Functions.. Dover: New York, 1958). ISBN 0-486-60462-4.
- 5
-
Cochran, W., Crick, F.H.C. and Vand V. 1952. The Structure of Synthetic Polypeptides. 1. The Transform of atoms on a helic. Acta Cryst. 5(5):581-586.
- 6
-
Crick, F.H.C. 1953a. The Fourier Transform of a Coiled-Coil., Acta Crystallographica 6(8-9):685-689.
- 7
-
Crick, F.H.C. 1953. The packing of
-helices- Simple coiled-coils.
Acta Crystallographica, 6(8-9):689-697.
- 8
-
Watson, J.D; Crick F.H.C. 1953a. Molecular Structure of Nucleic Acids - A Structure for Deoxyribose Nucleic Acid., Nature 171(4356):737-738.
- 9
- N. PISKUNOV: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. Kirjastus Valgus, Tallinn (1966).
- 10
- K. KURKI-SUONIO: Matemaattiset apuneuvot. Limes r.y., Helsinki (1966).
- 11
-
Watson, J.D; Crick F.H.C. 1953c. The Structure of DNA., Cold Spring Harbor Symposia on Qunatitative Biology 18:123-131.
- 12
-
I.S. Gradshteyn, I.M. Ryzhik, Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products., Academic Press, 2007.
ISBN 978-0-12-373637-6.
- 13
-
Spain,B., and M. G. Smith, Functions of mathematical physics., Van Nostrand Reinhold Company, London, 1970. Chapter 9: Bessel functions.
- 14
-
Abramowitz, M. and Stegun, I. A. (Eds.). Bessel Functions , Ch.9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358-364, 1972.
- 15
-
Arfken, G. Bessel Functions of the First Kind, and ``Orthogonality.'' Chs.11.1 and 11.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573-591 and 591-596, 1985.
- 16
-
Hansen, P. A. 1843. Ermittelung der absoluten Störungen in Ellipsen von beliebiger Excentricität und Neigung, I. Schriften der Sternwarte Seeberg. Gotha, 1843.
- 17
-
Lehmer, D. H. Arithmetical Periodicities of Bessel Functions. Ann. Math. 33, 143-150, 1932.
- 18
-
Le Lionnais, F. Les nombres remarquables (En: Remarcable numbers). Paris: Hermann, 1983.
- 19
-
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 619-622, 1953.
- 20
-
Schlömilch, O. X. 1857. Ueber die Bessel'schen Function. Z. für Math. u. Phys. 2: 137-165.
- 21
-
Spanier, J. and Oldham, K. B. "The Bessel Coefficients and " and "The Bessel Function ." Chs. 52-53 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 509-520 and 521-532, 1987.
- 22
-
Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.
- 23
-
Weisstein, Eric W. "Bessel Functions of the First Kind."
From MathWorld-A Wolfram Web Resource.
and
Graphs of Bessel Functions of the Second Kind
- 24
-
Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.
- 25
-
Watson, G. N. A Treatise on the Theory of Bessel Functions., (1995) Cambridge University Press. ISBN 0-521-48391-3.
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by pahio.