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# More Properties of Hankel and Bessel Functions

Plane waves, i.e. disturbances with planar wave fronts, can be subjected to translations in the Euclidean plane. They can also be used as basis functions for the two-dimensional Fourier transform. Both of these features extend to cylinder harmonics. The first one is captured by Property 19, the second one by Eq.(5.43) of Property 21. An example of a problem which uses the translation property for cylinder harmonics is a scattering problem similar to the one on page :

Consider a cylindrical source of waves and some distance away from it there is a scatterer also cylindrical in shape. Given the distance between these two cylinders, find the scattered wave field.

Property 19 (Addition theorem for cylinder harmonics)
A displaced cylinder harmonic is a linear superposition of the undisplaced cylinder harmonics. Mathematically one states this fact by the equation

 (534)

This equation is also known as the addition theorem'' for cylinder harmonics, be they singular or non-singular at the origin . The geometrical meaning of this theorem is as follows: Consider a displacement in the Euclidean plane by the vectorial amount and express this displacement in terms of polar coordinates:

Next, consider a point of observation, also expressed in terms of polar coordinates,

Finally, consider this same point of observation, but relative to the displaced origin at . In terms of polar coordinates one has
 (535)

where

are the observation coordinates relative to the displaced origin.

The problem is this: express a typical displaced cylinder harmonic,

a solution to the Helmholtz equation, in terms of the undisplaced cylinder harmonics,

 (536)

which are also solutions to the same Helmholtz equation.

The solution to this problem is given by the addition theorem'', Eq.(5.34).

It is interesting to note that both and , and hence are periodic functions of . Indeed, one notices that

or, equivalently, that

 As a consequence, the old and the new polar coordinates are related by and

Thus one is confronted with the problem of finding the Fourier series of the periodic function

The solution to this problem is given by the addition theorem'', Eq.(5.34). We shall refrain from validating this Fourier series by a frontal assault. Instead, we give a simple three-step geometrical argument. It accomplishes the task of expressing the displaced cylinder harmonics in terms of the undisplaced cylinder harmonics

(i)
Represent the displaced harmonic as a linear combination of plane waves in the usual way

 (537)

(ii)
take each of these plane waves and reexpress them relative to the undisplaced origin:

The phase shift factor is a plane wave amplitude in its own right, which depends periodically on the angel , and is therefore, according to Property 18, a linear combination of Bessel harmonics

(iii)
Reintroduce the translated plane wave

and its concomitant phase shift factor from step (ii) into the displaced cylinder harmonic. The result is a linear sum of phase shifted cylinder harmonics, Eq.(5.37),

According to the definitions, Eqs.(5.4)-(5.5), the integral is a cylinder harmonic of order . Consequently, one obtains

Multiplying both sides by yields the following geometrically perspicuous result:

Note that the left hand side is a displaced cylinder harmonic of order relative to the new -axis which point along the displacement vector and whose origin lies along the tip of this vector. The angle is the new angle of observation relative to the new tilted -axis and the new origin.

The sum on the right is composed of the cylinder harmonics of order undisplaced relative to the tilted -axis. The angle is the old angle of observation relative to the tilted -axis and the old origin.

The displacement formula can be summarized as follows

Property 20 (Translations represented by cylinder harmonics)
It is amusing to specialize to the case where is an integer and is a Bessel function of integral order . In that case the displacement formula becomes

or equivalently, after changing the summation index,
 (538)

where

while Eq.(5.35) for the vector triangle becomes

 (539)

Compare Eq.(5.39) with Eq.(5.38). Observe that (i) for each translation in the Euclidean plane, say , there is a corresponding infinite dimensional matrix

and (ii) the result of successive translations, such as Eq.(5.39), is represented by the product of the corresponding matrices, Eq.(5.38).

Exercise 54.1 (ADDITION FORMULA FOR BESSEL FUNCTIONS)
Express as a sum of products of Bessel functions of and respectively.

Property 21 (Completeness)
The cylinder waves form a complete set. More precisely,
 (540)

This relation is the cylindrical analogue of the familiar completeness relation for plane waves,
 (541)

In fact, the one for plane waves is equivalent to the one for cylinder waves. The connecting link between the two is the plane wave expansion, Eq.(5.24),

Introduce it into Eq.(5.41) and obtain

Using the orthogonality property
 (542)

the definition

and

one obtains
 (543)

the completeness relation for the cylinder waves.

Property 22 (Fourier-Bessel transform)
The Bessel functions of fixed integral order form a complete set
 (544)

This result is a direct consequence of Property 21. Indeed, multiply the cylinder wave completeness relation, Eq.(5.40) by , integrate over from 0 to , again use the orthogonality property, Eq. 5.42, and cancel out the factor common factor from both sides. The result is Eq.(5.44), the completeness relation for the Bessel functions on the positive -axis.

Remark: By interchanging the roles of and one obtain from Eq.(5.44)

Remark: The completeness relation, Eq.(5.44), yields

where

This is the Fourier-Bessel transform theorem.

It is interesting to note that the completeness relation, Eq.(5.44), is independent of the integral order of . One therefore wonders whether Eq.(5.44) also holds true if one uses , Bessel functions of any complex order . This is ideed the case.

Property 23 (Bessel transform)
The Bessel functions of complex order form a complete set
 (545)

This result gives rise to the transform pair
 (546) (547)

and it is obvious that mathematically Property 22 is a special case of Property 23.

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Ulrich Gerlach 2010-12-09