Properties of Hankel and Bessel Functions

Associated with the two kinds of Hankel functions are two solutions to the
Helmholtz equation. They are the ``cylinder harmonics'' or order
,

and

Here

are normalization constants whose values are derived below (see Property 11 below).

The name ``cylinder harmonic'' arises from the fact that these two functions emerge from those solutions of the Helmholtz equation whose level surfaces mold themselves naturally to the cylindrical geometry of its domain. These functions have the following properties:

They are linear superpositions of plane waves.

The integral representatives

of the two Hankel functions do not depend on any real changes in the integration limits.

This means that the -dependent shift in the limits of the integral has no effect on the value of the integral itself, whenever the integration limits and each lie near infinity in a strip of convergence of the integral.

Suppose the integration contour is taken to be the curve labelled , where is near the vertical line and is near . Then for we see that

This equality is a result of two facts:

- The dominant contribution comes from the path between and , and that path can be deformed into the original curve and by the Cauchy-Goursat theorem the integral will remain unchanged.
- The -dependent change due to the shift in the end points and is neglegible, because the integrand is already neglegibly small at these points.

To summarize:

represents a continuous function of whenever . When this inequality is fulfilled one has

If this inequality is not fulfilled, then the right hand side of this equation diverges and is therefore not valid. This is because in this case the contour of the integral on r.h.s. of Eq.(5.6) cannot be deformed into that of the l.h.s.: The integration limits would fall outside the shaded strips of convergence, the integral would diverge, and the r.h.s. of Eq.(5.7) would loose its meaning.

As usual, the circumstance or are defined in terms of limits as from the inside of the interval..

Thus we conclude that and are independent of indeed. The result is that the two cylinder harmonics have the form

a product of two functions, each one depending separately on its own variable, but independent of the other.

The cylinder harmonics are eigenfunctions of the rotation generator ,

that is to say, they are invariant (modulo constant multiplicative factor) under rotation around the origin

They satisfy the Helmholtz's equation, which in polar coordinates becomes Bessel's equation

0 | |||

Even through the eigenvalue of the operator is infinitely degenerate, the eigenvalues of in the equation

serve to distinguish the elements of the degenerate set.

The domain of a

which satisfies the wave equation

whenever the constants , and satisfy the dispersion relation

The dependent factors of these cylinder harmonics,

are called

It is worthwhile to reemphasize that the integral representations of
and
converge and are well defined for
*any complex number*
.

Having equal normalization constants,

the two Hankel functions, Eqs. (5.4) and (5.5), determine the

One arrives at this definition
by means of the *union* of the two paths
and
which define
and
. By the Cauchy-Goursat theorem
these paths can be deformed into a *single* path as depicted in
Figure 5.7

Its integral representation requires the two integration contours depicted in Figure 5.6.

The Hankel functions are the analogues of the

for large , as we shall see later.

The next property asks and answers the the question: How do the Bessel and the Neuman functions depend on their complex order ? With the universally agreed-upon value for the normalization constant , the answer could be no simpler: For real these functions are real and for complex these functions are their analytic extensions into the complex domain. More precisely, we have

- If the
*order*is*real*then their sum (the ``Bessel function'')-
is
*real*when is real and

*provided*the normalization constant is -
is
- If
is
*complex*, then, for fixed positive , both and are analytic functions of their order . Furthermore, they obey the*reflection principle:*

The reflection principle is a general property which analytic functions enjoy whenever their values are real on the real ( ) axis. It is shown below that the form of the normalization constant guarantees this. Indeed, for the Bessel function the proof consists of three steps below. (We delay the application of the reflection principle to the Neumann function until after we have exhibited the complex conjugation property applied to the two Hankel functions on page .)

*Step 1:* Deform the integration path into straight lines.
The result is

*Step 2:* Symmetrize the integrals by shifting the integration
limits to the left. This is achieved by introducing the new dummy variable

The result is

Reminder: We have not shifted the path of integration. Instead, we have only altered the coordinate labelling used to describe that path in the complex plane.

*Step 3:*

- Fix the normalization constant by requiring that be real when is real. This is achieved by setting . This cancels out the last factor.
- To bring this reality of
to light, combine the first and
third integral by introducing

*Conclusion:* When
is real, then

also, if is complex, then by inpection one finds

which is what we set out to show.

The Bessel functions of integral order is given by

This equation is the result of changing the integration variable . Letting , one obtains

Show that

The Bessel function of complex order and for real has the following Frobenious expansion around the origin

The power series, together with its normalization constant, follows from the integral representation

where is the integration contour indicated in Figure 5.7. Indeed, introduce the new variable of integration

Under this change, the new integration contour is the one depicted in Figure 5.8, which is based on the following scheme:

The integral becomes

By expanding the exponential in a Taylor series one obtains Eq.(5.11), provided one sets

which is one of the definitions of the gamma function. This contour integral is meaningless unless one specifies the branch of the multiple-valued function . The branch is dictated by the requirement that

For this branch the domain is restricted to whenever the cut for this branch is the positive -axis, as in Figure 5.8.

Let us look at the solutions to Bessel's equation from the viewpoint of linear algebra. The solution space is two dimensional. There are two important spanning sets. The first one,

is simple whenever . By contrast, the second one

is simple whenever . However, we know that these two bases are related by a linear transformation.

The question is: what is this linear transformation? The answer is provided by the following

When the order of a Bessel function is not an integer ( ), then the set of Bessel functions form an independent set. Moreover, one has

If the Hankel functions are of real order (and ), then

i.e., they are complex conjugates of each other. This follows from equations 5.12 and 5.13 of Property 13.

*Remark:* There are three additional consequences:

First of all, it follows from Property 12 that if is complex, then

Second, apply this complex conjugation property to the defining Eq.(5.9) and obtain the reflection principle applied to the Neumann functions

Third, if is complex also, then

Returning to the validation of the Hankel-Bessel identities, one finds that this process consists of four steps. They consist primarily of manipulating the intergration paths of the integral representations of and .

*Step 1.* Recall the definition of
:

Here the is the complex conjugate of the integration path

The rest of the proof consists of

*Step 2.* Subtract this from an analogous expression for
.

*Step 3.* Deform the contour and reexpress the r.h.s. in
terms of
. This yields the desired equation.

*Step 4.* Use Property 8 to obtain the expression for
.

The details of these remaining steps are

*Step 2.* Subtract the expression
in Step 1 from the analogous expression for
. After a slight
deformation of the two respective integration path, obtain

Here is the integration contour for and is shifted by to the left.

*Step 3.*

- Recall from Property 7 that
- In addition, we have

- Introduce the results of 1. and 2. into the last expression in Step 2,
and obtain

*Step 4.* Use Property 8 to obtain

These are the two expressions for the two kinds of Hankel functions in terms of Bessel functions of order and .

Let or or or be any solution to Bessel's equation of

*Step 1.* Apply the definition to the sum and difference

*Step 2.* Observe that

Consequently,

These are the two recursion relations (5.14) and (5.15).

These relations are quite useful. Note that
by adding and subtracting the recursion relations one obtains

Let us call

and

These operators yield

and

which is the (rotationally and translation) invariant Laplacian operator.

**Comment:** (*Factorization Method for Finding
Cylinder Harmonics.*)

It is difficult to exclude these raising and lowering operators as the fastest way for establishing relationships between normal modes in a cylindrical cavity. For example, suppose one knows explicitly the rotationally symmetric mode . Then all the other modes

can be obtained by repeated application of the raising operator . The lowering operator undoes the work of the raising operators

i.e.,

This feature also illustrates the fact that the -dimensional

Recalling the definition of the rotation generator in Section 5.1.4, notice that

This commutation relation is fundamental for the following reason:

Suppose we have a solution to the Helmholtz equation

and suppose that the solution is a

Then the commutation relation implies

In other words, is another rotation eigenfunction. Furthermore,

i.e. the new rotation eigenfunction

is again a solution to the Helmholtz equation. The analogous result holds for .

*To summarize:* The algebraic method for solving the Helmholtz
equation is a two step process: (i) Factor the Laplacian,
Eq.(5.1) into two factors
and
, and
(ii) for each eigenspace of
construct a basis using
and
, whose capability as raising and lowering operators
is implied by the two commutation relations

These operators obviously commute,

This is evident from Eq.(5.16). Furthermore, as we have seen from Eq.(5.17), the fact that

acts as a guarantee that all the basis elements obtained from these raising and lowering operators lie in the same subspace characterized by the degenerate eigenvalue of the Laplacian.

For illustrative purposes we compute
the first few cylinder harmonics. Starting with
, one obtains:

(521) | |||

(522) |

Recall that the cylinder harmonics of (complex) order where constructed as a linear superposition of plane wave solutions

and (ii) the requirement of being finite at the origin . These boundary conditions give rise to the cylinder waves

We see that this is the th Fourier coefficient of in disguise:

This represents a plane wave propagating along the axis, and it is easy to remember.

Indeed, if one replaces
with
in Eq.(5.23), one obtains

This means that any plane wave in the Euclidean plane can be represented as a linear combination of cylinder harmonics of integral order.