Bessel's Special Function - 3 | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

HANKEL FUNCTIONS

Many authors prefer to introduce the Hankel functions by means of integral representations and then to use them to define the Neumann function N_ν(z). An outline of this approach is given at the end of this section.

Definitions

Because we have already obtained the Neumann function by more elementary (and less powerful) techniques, we may use it to define the Hankel functions H_ν⁽¹⁾ (x ) and H_ν⁽²⁾ (x ):

(11.85)

and

 (11.86)

This is exactly analogous to taking

Since the Hankel functions are linear combinations (with constant coefficients) of J_ν and N_ν , they satisfy the same recurrence relations (Eqs. (11.10) and (11.12))

Example 11.4.1 CYLINDRICAL TRAVELING WAVES 

As an illustration of the use of Hankel functions, consider a two-dimensional wave problem similar to the vibrating circular membrane of Exercise 11.1.25. Now imagine that the waves are generated at r = 0 and move outward to infinity. We replace our standing waves by traveling ones. The differential equation remains the same, but the boundary conditions change. We now demand that for large r the wave behave like

              (11.97)

to describe an outgoing wave. As before, k is the wave number. This assumes, for simplicity, that there is no azimuthal dependence, that is, no angular momentum, or m = 0. In Sections 7.3 and 11.6, H⁽¹⁾₀ (kr ) is shown to have the asymptotic behavior (for r →∞)

         (11.98)

This boundary condition at infinity then determines our wave solution as

               (11.99)

This solution diverges as r → 0, which is the behavior to be expected with a source at the origin.

The choice of a two-dimensional wave problem to illustrate the Hankel function H₀⁽¹⁾ (z) is not accidental. Bessel functions may appear in a variety of ways, such as in the separation of conical coordinates. However, they enter most commonly in the radial equations from the separation of variables in the Helmholtz equation in cylindrical and in spherical polar coordinates. We have taken a degenerate form of cylindrical coordinates for this illustration. Had we used spherical polar coordinates (spherical waves), we should have encountered index  ,n an integer. 

Contour Integral Representation of the Hankel Functions 

The integral representation (Schlaefli integral)

may easily be established as a Cauchy integral for ν = n, an integer (by recognizing that the numerator is the generating function (Eq. (11.1)) and integrating around the origin).
If ν is not an integer, the integrand is not single-valued and a cut line is needed in our complex plane. Choosing the negative real axis as the cut line and using the contour shown in Fig. 11.7, we can extend Eq. (11.100) to nonintegral ν . Substituting Eq. (11.100) into Bessel’s ODE, we can represent the combined integrand by an exact differential that vanishes as t →∞e^±iπ .
We now deform the contour so that it approaches the origin along the positive real axis, as shown in Fig. 11.8. For x> 0, this particular approach guarantees that the exact differential mentioned will vanish as t → 0 because of the e^−x/2t → 0 factor. Hence each of the separate portions (∞ e^−iπ to 0) and (0 to ∞ e^iπ ) is a solution of Bessel’s equation. We define

These expressions are particularly convenient because they may be handled by the method of steepest descents. H_ν⁽¹⁾ (x ) has a saddle point at t =+i , whereas H_ν⁽²⁾ (x ) has a saddle point at t =−i .

FIGURE 11.7 Bessel function contour.

The problem of relating Eqs. (11.101) and (11.102) to our earlier definition of the Hankel function (Eqs. (11.85) and (11.86)) remains. Since Eqs. (11.100) to (11.102) combined yield

                 (11.103)

by inspection, we need only show that

              (11.104)

This may be accomplished by the following steps :

Integral representations have appeared before: Eq. (8.35) for  and various representations of J_ν (z). With these integral representations of the Hankel functions, it is perhaps appropriate to ask why we are interested in integral representations. There are at least four reasons. The first is simply aesthetic appeal. Second, the integral representations help to distinguish between two linearly independent solutions. In Fig. 11.6, the contours C₁ and C₂ cross different saddle points. For the Legendre functions the contour for P_n (z) (Fig. 12.11) and that for Q_n (z) encircle different singular points.

Third, the integral representations facilitate manipulations, analysis, and the development of relations among the various special functions. Fourth, and probably most important of all, the integral representations are extremely useful in developing asymptotic expansions. One approach, the method of steepest descents. A second approach, the direct expansion of an integral representation for the modified Bessel function K_ν (z). This same technique may be used to obtain asymptotic expansions of the confluent hypergeometric functions M and U .
In conclusion, the Hankel functions are introduced here for the following reasons:

• As analogs of e^±ix they are useful for describing traveling waves.
• They offer an alternate (contour integral) and a rather elegant definition of Bessel functions.
• H_ν⁽¹⁾ is used to define the modified Bessel function K_ν.

MODIFIED BESSEL FUNCTIONS, I_ν (x ) AND K_ν (x )

The Helmholtz equation,

The analog to Eq. (11.22a) is

The Helmholtz equation may be transformed into the diffusion equation by the transformation k → ik . Similarly, k → ik changes Eq. (11.22a) into Eq. (11.109) and shows that

Y_ν (kρ ) = Z_ν (i kρ ).

The solutions of Eq. (11.109) are Bessel functions of imaginary argument. To obtain a solution that is regular at the origin, we take Z_ν as the regular Bessel function J_ν .Itis customary (and convenient) to choose the normalization so that

               (11.110)

(Here the variable kρ is being replaced by x for simplicity.) The extra i ^−ν normalization cancels the i ^ν from each term and leaves I_ν (x ) real. Often this is written as

                 (11.111)

I₀ and I₁ are shown in Fig. 11.10.

Series Form 

In terms of infinite series this is equivalent to removing the (−1)^s sign in Eq. (11.5) and writing

 Recurrence Relations 

The recurrence relations satisfied by I_ν (x ) may be developed from the series expansions, but it is perhaps easier to work from the existing recurrence relations for J_ν (x ). Let us replace x by −ix and rewrite Eq. (11.110) as

            (11.114)

Then Eq. (11.10) becomes

Replacing x by ix , we have a recurrence relation for I_ν (x ),

        (11.115)

Equation (11.12) transforms to

             (11.116)

These are the recurrence relations. It is worth emphasizing that although two recurrence relations, Eqs. (11.115) and (11.116) or  specify the second-order ODE, the converse is not true. The ODE does not uniquely fix the recurrence relations. Equations (11.115) and (11.116)  provide an example.
From Eq. (11.113) it is seen that we have but one independent solution when ν is an integer, exactly as in the Bessel functions J_ν . The choice of a second, independent solution of Eq. (11.108) is essentially a matter of convenience. The second solution given here is selected on the basis of its asymptotic behavior — as shown in the next section. The confusion of choice and notation for this solution is perhaps greater than anywhere else in this field.  Many authors choose to define a second solution in terms of the Hankel function H_ν⁽¹⁾ (x ) by

analogous to Eq. (11.60) for N_ν (x ). The choice of Eq. (11.117) as a definition is somewhat unfortunate in that the function K_ν (x ) does not satisfy the same recurrence relations as I_ν (x ) . zTo avoid this annoyance, other authors²¹ have included an additional factor of cos νπ . This permits K_ν to satisfy the same recurrence relations as I_ν , but it has the disadvantage of making K_ν = 0for

The series expansion of K_ν (x ) follows directly from the series form of H_ν⁽¹⁾ (i x ).The lowest-order terms are (cf. Eqs. (11.61) and (11.62))

Because the modified Bessel function I_ν is related to the Bessel function J_ν , much as sinh is related to sine, I_ν and the second solution K_ν are sometimes referred to as hyperbolic Bessel functions. K₀ and K₁ are shown in Fig. 11.10.
I₀ (x ) and K₀ (x ) have the integral representations

Equation (11.120) may be derived from Eq. (11.30) for J₀ (x ) or may be taken as  ν = 0. The integral representation of K₀ , Eq. (11.121), is a Fourier transform and may best be derived with Fourier transforms, Chapter 15, or with Green’s functions. A variety of other forms of integral representations (including ν ≠ 0) appear in the exercises. These integral representations are useful in developing asymptotic forms  and in connection with Fourier transforms.
To put the modified Bessel functions I_ν (x ) and K_ν (x ) in proper perspective, we introduce them here because:

•  These functions are solutions of the frequently encountered modified Bessel equation.
• They are needed for specific physical problems, such as diffusion problems.
• K_ν (x ) provides a Green’s function.
• K_ν (x ) leads to a convenient determination of asymptotic behavior .