Bessel functions and helical structure diffraction patterns

Bessel functions and helical structure diffraction patterns

1

The Bessel functions and Helical Structure

Diffraction represented by Bessel functions.

The

linear differential equation

x

2

⁢

d

2

⁢

y

d

⁢

x

2

+

x

⁢

d

⁢

y

d

⁢

x

+

(

x

2

-

p

2

)

⁢

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

=

x

r

⁢

∑

k

=

0

∞

a

k

⁢

x

k

=

∑

k

=

0

∞

a

k

⁢

x

r

+

k

,

(1.2)

due to Frobenius.  Since the

parameter

r

is

indefinite

, we may regard

a

0

as distinct from 0.

We substitute (2) and the

derivatives

of the series in (1):

x

2

⁢

∑

k

=

0

∞

(

r

+

k

)

⁢

(

r

+

k

-

1

)

⁢

a

k

⁢

x

r

+

k

-

2

+

x

⁢

∑

k

=

0

∞

(

r

+

k

)

⁢

a

k

⁢

x

r

+

k

-

1

+

(

x

2

-

p

2

)

⁢

∑

k

=

0

∞

a

k

⁢

x

r

+

k

=

0

.

Thus the

coefficients

of the powers

x

r

,

x

r

+

1

,

x

r

+

2

and so on must vanish, and we get the system of equations

{

[

r

2

-

p

2

]

⁢

a

0

=

0

,

[

(

r

+

1

)

2

-

p

2

]

⁢

a

1

=

0

,

[

(

r

+

2

)

2

-

p

2

]

⁢

a

2

+

a

0

=

0

,

…

[

(

r

+

k

)

2

-

p

2

]

⁢

a

k

+

a

k

-

2

=

0

.

(1.3)

The last of those can be written

(

r

+

k

-

p

)

⁢

(

r

+

k

+

p

)

⁢

a

k

+

a

k

-

2

=

0

.

Because

a

0

≠

0

,  the first of those (the

indicial equation

) gives

r

2

-

p

2

=

0

,  i.e. we have the roots

r

1

=

p

,

r

2

=

-

p

.

Let’s first look the the solution of (1) with

r

=

p

;  then

k

⁢

(

2

⁢

p

+

k

)

⁢

a

k

+

a

k

-

2

=

0

,  and thus

a

k

=

-

a

k

-

2

k

(

2

p

+

k

)

.

From the system (3) we can solve one by one each of the coefficients

a

1

,

a

2

,

…

and express them with

a

0

which remains arbitrary.  Setting for

k

the integer values we get

{

a

1

=

0

,

a

3

=

0

,

…

,

a

2

⁢

m

-

1

=

0

;

a

2

=

-

a

0

2

⁢

(

2

⁢

p

+

2

)

,

a

4

=

a

0

2

⋅

4

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

,

…

,

a

2

⁢

m

=

(

-

1

)

m

⁢

a

0

2

⋅

4

⋅

6

⁢

⋯

⁢

(

2

⁢

m

)

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

⁢

…

⁢

(

2

⁢

p

+

2

⁢

m

)

(1.4)

(where

m

=

1

,

2

,

…

).
Putting the obtained coefficients to (2) we get the particular solution

y

1

:=

a

0

x

p

[

1

-

x

2

2

⁢

(

2

⁢

p

+

2

)

+

x

4

2

⋅

4

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

-

x

6

2

⋅

4

⋅

6

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

⁢

(

2

⁢

p

+

6

)

+

-

…

]

(1.5)

In order to get the coefficients

a

k

for the second root

r

2

=

-

p

we have to look after that

(

r

2

+

k

)

2

-

p

2

≠

0

,

or

r

2

+

k

≠

p

=

r

1

.  Therefore

r

1

-

r

2

=

2

⁢

p

≠

k

where

k

is a

positive integer

.  Thus, when

p

is not an

integer

and not an integer added by

1

2

, we get the second particular solution, gotten of (5) by replacing

p

by

-

p

:

y

2

:=

a

0

x

-

p

[

1

-

x

2

2

⁢

(

-

2

⁢

p

+

2

)

+

x

4

2

⋅

4

⁢

(

-

2

⁢

p

+

2

)

⁢

(

-

2

⁢

p

+

4

)

-

x

6

2

⋅

4

⋅

6

⁢

(

-

2

⁢

p

+

2

)

⁢

(

-

2

⁢

p

+

4

)

⁢

(

-

2

⁢

p

+

6

)

+

-

…

]

(1.6)

The

power series

of (5) and (6) converge for all values of

x

and are

linearly independent

(the ratio

y

1

/

y

2

tends to 0 as

x

→

∞

).  With the appointed value

a

0

=

1

2

p

⁢

Γ

⁢

(

p

+

1

)

,

the solution

y

1

is called the

Bessel function

of the

first kind

and of order

p

and denoted by

J

p

.  The

similar

definition

is set for the first kind Bessel function of an arbitrary order

p

∈

ℝ

(and

ℂ

).
For

p

∉

ℤ

the general solution of the Bessel’s

differential equation

is thus

y

:=

C

1

⁢

J

p

⁢

(

x

)

+

C

2

⁢

J

-

p

⁢

(

x

)

,

where

J

-

p

⁢

(

x

)

=

y

2

with

a

0

=

1

2

-

p

⁢

Γ

⁢

(

-

p

+

1

)

.

The explicit

expressions

for

J

±

p

are

J

±

p

⁢

(

x

)

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

Γ

⁢

(

m

±

p

+

1

)

⁢

(

x

2

)

2

⁢

m

±

p

,

(1.7)

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

gamma function

.

E.g. when

p

=

1

2

the series in (5) gets the form

y

1

=

x

1

2

2

⁢

Γ

⁢

(

3

2

)

[

1

-

x

2

2

⋅

3

+

x

4

2

⋅

4

⋅

3

⋅

5

-

x

6

2

⋅

4

⋅

6

⋅

3

⋅

5

⋅

7

+

-

…

]

=

2

π

⁢

x

(

x

-

x

3

3

!

+

x

5

5

!

-

+

…

)

.

Thus we get

J

1

2

⁢

(

x

)

=

2

π

⁢

x

⁢

sin

⁡

x

;

analogically (6) yields

J

-

1

2

⁢

(

x

)

=

2

π

⁢

x

⁢

cos

⁡

x

,

and the general solution of the equation (1) for

p

=

1

2

is

y

:=

C

1

⁢

J

1

2

⁢

(

x

)

+

C

2

⁢

J

-

1

2

⁢

(

x

)

.

In the case that

p

is a non-negative integer

n

, the “+” case of (7) gives the solution

J

n

⁢

(

x

)

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

(

m

+

n

)

!

⁢

(

x

2

)

2

⁢

m

+

n

,

but for

p

=

-

n

the expression of

J

-

n

⁢

(

x

)

is

(

-

1

)

n

⁢

J

n

⁢

(

x

)

, i.e. linearly dependent of

J

n

⁢

(

x

)

.  It can be shown that the other solution of (1) ought to be searched in the form

y

=

K

n

⁢

(

x

)

=

J

n

⁢

(

x

)

⁢

ln

⁡

x

+

x

-

n

⁢

∑

k

=

0

∞

b

k

⁢

x

k

.  Then the general solution is

y

:=

C

1

⁢

J

n

⁢

(

x

)

+

C

2

⁢

K

n

⁢

(

x

)

.

Other formulae

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

F

:

F

⁢

(

z

,

t

)

=

e

z

2

⁢

(

t

-

1

t

)

=

∑

n

=

-

∞

∞

J

n

⁢

(

z

)

⁢

t

n

(1.8)

This

function

has an

essential singularity

at

t

=

0

but is

analytic

elsewhere in

ℂ

; thus

F

has the

Laurent expansion

in that point.  Let us prove (8) by using the general expression

c

n

=

1

2

⁢

π

⁢

i

⁢

∮

γ

f

⁢

(

t

)

(

t

-

a

)

n

+

1

⁢

𝑑

t

of the

coefficients of Laurent series

.  Setting to this

a

:=

0

,

f

⁢

(

t

)

:=

e

z

2

⁢

(

t

-

1

t

)

,

ζ

:=

z

⁢

t

2

gives

c

n

=

1

2

⁢

π

⁢

i

⁢

∮

γ

e

z

⁢

t

2

⁢

e

-

z

2

⁢

t

t

n

+

1

⁢

𝑑

t

=

1

2

⁢

π

⁢

i

⁢

(

z

2

)

n

⁢

∮

δ

e

ζ

⁢

e

-

z

2

4

⁢

ζ

ζ

n

+

1

⁢

𝑑

ζ

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

(

z

2

)

2

⁢

m

+

n

⁢

1

2

⁢

π

⁢

i

⁢

∮

δ

ζ

-

m

-

n

-

1

⁢

e

ζ

⁢

𝑑

ζ

.

The paths

γ

and

δ

go once round the origin anticlockwise in the

t

-plane and

ζ

-plane, respectively.  Since the

residue

of

ζ

-

m

-

n

-

1

⁢

e

ζ

in the origin is

1

(

m

+

n

)

!

=

1

Γ

⁢

(

m

+

n

+

1

)

,  the

residue theorem

(

http://planetmath.org/CauchyResidueTheorem

) gives

c

n

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

Γ

⁢

(

m

+

n

+

1

)

⁢

(

z

2

)

2

⁢

m

+

n

=

J

n

⁢

(

z

)

.

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:

J

n

⁢

(

z

)

=

1

π

⁢

∫

0

π

cos

⁡

(

n

⁢

φ

-

z

⁢

sin

⁡

φ

)

⁢

𝑑

φ

Also one can obtain the

addition formula

J

n

⁢

(

x

+

y

)

=

∑

ν

=

-

∞

∞

J

ν

⁢

(

x

)

⁢

J

n

-

ν

⁢

(

y

)

and the series of cosine and sine:

cos

z

=

J

0

(

z

)

-

2

J

2

(

z

)

+

2

J

4

(

z

)

-

+

…

sin

z

=

2

J

1

(

z

)

-

2

J

3

(

z

)

+

2

J

5

(

z

)

-

+

…

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 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 (

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.

the diffraction

pattern

of a helical molecule wrapped around a

cylinder

computed from the

Fourier transform

of the helix in

cylindrical coordinates

;

2.

electromagnetic waves in a cylindrical waveguide

3.

diffusion problems on a

lattice

.

4.

vibration modes of a thin

circular

, tubular or annular membrane (such as a drum, other membranophone, the vocal cords, etc.)

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

-

h

⁢

e

⁢

l

⁢

i

⁢

x

) 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.

References

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 Fractions

. New York: Chelsea, 1948.

23

Weisstein, Eric W. ”Bessel Functions of the First Kind.”

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

From MathWorld–A Wolfram Web Resource. and

http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html

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.

.2

Linked files PDFs:

http://aux.planetmath.org/files/objects//BesselFunctions-DiffractionHelices.pdf
http://aux.pp-dev.org:8080/files/objects/716/BesselFunctions_AppsToDiffraction_HelicalStructures.pdf

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 functions

Defines

Bessel functions

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