Bessel functions and helical structure diffraction patterns

1 The Bessel functions and Helical Structure
Diffraction represented by Bessel functions.

The linear differential equation

x2d2ydx2+xdydx+(x2-p2)y=0, (1.1)

in which p 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

y=xrk=0akxk=k=0akxr+k, (1.2)

due to Frobenius.  Since the parameterMathworldPlanetmath r is indefinite, we may regard a0 as distinct from 0.

We substitute (2) and the derivatives of the series in (1):


Thus the coefficients of the powers xr, xr+1, xr+2 and so on must vanish, and we get the system of equations

{[r2-p2]a0=0,[(r+1)2-p2]a1=0,[(r+2)2-p2]a2+a0=0,[(r+k)2-p2]ak+ak-2=0. (1.3)

The last of those can be written


Because  a00,  the first of those (the indicial equationMathworldPlanetmath) gives  r2-p2=0,  i.e. we have the roots


Let’s first look the the solution of (1) with  r=p;  then  k(2p+k)ak+ak-2=0,  and thus


From the system (3) we can solve one by one each of the coefficients a1, a2,   and express them with a0 which remains arbitrary.  Setting for k the integer values we get

{a1=0,a3=0,,a2m-1=0;a2=-a02(2p+2),a4=a024(2p+2)(2p+4),,a2m=(-1)ma0246(2m)(2p+2)(2p+4)(2p+2m) (1.4)

(where  m=1, 2,). Putting the obtained coefficients to (2) we get the particular solution

y1:=a0xp[1-x22(2p+2)+x424(2p+2)(2p+4)-x6246(2p+2)(2p+4)(2p+6)+-] (1.5)

In order to get the coefficients ak for the second root  r2=-p  we have to look after that


or  r2+kp=r1.  Therefore


where k is a positive integer.  Thus, when p is not an integer and not an integer added by 12, we get the second particular solution, gotten of (5) by replacing p by -p:

y2:=a0x-p[1-x22(-2p+2)+x424(-2p+2)(-2p+4)-x6246(-2p+2)(-2p+4)(-2p+6)+-] (1.6)

The power seriesMathworldPlanetmath of (5) and (6) converge for all values of x and are linearly independentMathworldPlanetmath (the ratio y1/y2 tends to 0 as  x).  With the appointed value


the solution y1 is called the Bessel functionDlmfMathworldPlanetmathPlanetmath of the first kind and of order p and denoted by Jp.  The similarMathworldPlanetmathPlanetmath definition is set for the first kind Bessel function of an arbitrary order  p (and ). For  p  the general solution of the Bessel’s differential equation is thus


where  J-p(x)=y2  with  a0=12-pΓ(-p+1).

The explicit expressions for J±p are

J±p(x)=m=0(-1)mm!Γ(m±p+1)(x2)2m±p, (1.7)

which are obtained from (5) and (6) by using the last for gamma functionDlmfDlmfMathworldPlanetmath.

E.g. when  p=12  the series in (5) gets the form


Thus we get


analogically (6) yields


and the general solution of the equation (1) for  p=12  is


In the case that p is a non-negative integer n, the “+” case of (7) gives the solution


but for  p=-n  the expression of J-n(x) is (-1)nJn(x), i.e. linearly dependent of Jn(x).  It can be shown that the other solution of (1) ought to be searched in the form  y=Kn(x)=Jn(x)lnx+x-nk=0bkxk.  Then the general solution is  y:=C1Jn(x)+C2Kn(x).

Other formulae

The first kind Bessel functions of integer order have the generating function F:

F(z,t)=ez2(t-1t)=n=-Jn(z)tn (1.8)

This functionMathworldPlanetmath has an essential singularity at  t=0  but is analytic elsewhere in ; thus F 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  a:=0,  f(t):=ez2(t-1t),  ζ:=zt2  gives


The paths γ and δ go once round the origin anticlockwise in the t-plane and ζ-plane, respectively.  Since the residueDlmfMathworldPlanetmath of ζ-m-n-1eζ in the origin is  1(m+n)!=1Γ(m+n+1),  the residue theoremMathworldPlanetmath ( gives


This that F has the Laurent expansion (8).

By using the generating function, one can easily derive other formulae, e.g. the of the Bessel functions of integer order:


Also one can obtain the addition formulaPlanetmathPlanetmath


and the series of cosine and sine:


2 Applications of Bessel functions in Physics and Engineering

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 coordinatesMathworldPlanetmath. 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 (α=n), whereas in spherical coordinates, such solutions involve Bessel functions of half-integer orders (α=n+1/2). Several examples of Bessel function solutions are:

  1. 1.

    the diffraction pattern of a helical molecule wrapped around a cylinder computed from the Fourier transform of the helix in cylindrical coordinatesMathworldPlanetmath;

  2. 2.

    electromagnetic waves in a cylindrical waveguide

  3. 3.

    diffusion problems on a lattice.

  4. 4.

    vibration modes of a thin circular, tubular or annular membrane (such as a drum, other membranophone, the vocal cords, etc.)

  5. 5.

    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.

2.1 Applications of Bessel functions in Physical Crystallography

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. α-helix) 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 α-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 Strungen in Ellipsen von beliebiger Excentricitat 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 FractionsDlmfMathworldPlanetmath. New York: Chelsea, 1948.
  • 23 Weisstein, Eric W. ”Bessel Functions of the First Kind.” MathWorld–A Wolfram Web Resource. and 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.

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Title Bessel functions and helical structure diffraction patterns
Canonical name BesselFunctionsAndHelicalStructureDiffractionPatterns
Date of creation 2013-03-22 19:23:04
Last modified on 2013-03-22 19:23:04
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 6
Author bci1 (20947)
Entry type Topic
Classification msc 33C10
Classification msc 78A45
Classification msc 00A79
Related topic BesselFunction
Related topic BesselsEquation
Defines helical structure diffraction patterns
Defines cylinder functionsDlmfMathworld
Defines Bessel functions