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**Bessel Functions of Nonintegral Order **

These different approaches are not exactly equivalent. The generating function approach is very convenient for deriving two recurrence relations, Bessel’s differential equation, integral representations, addition theorems, and upper and lower bounds. However, you will probably have noticed that the generating function deﬁned only Bessel functions of integral order, J_{0} ,J_{1} ,J_{2} , and so on. This is a limitation of the generating function approach that can be avoided by using the contour integral instead, thus leading to foregoing approach (3). But the Bessel function of the ﬁrst kind, J_{ν }(x ), may easily be deﬁned for nonintegral ν by using the series (Eq. (11.5)) as a new deﬁnition.

The recurrence relations may be veriﬁed by substituting in the series form of J_{ν} (x ). From these relations Bessel’s equation follows. In fact, if ν is not an integer, there is actually an important simpliﬁcation. It is found that Jν and J_{−ν} are independent, for no relation of the form of Eq. (11.8) exists. On the other hand, for ν = n, an integer, we need another solution.

**ORTHOGONALITY**

If Bessel’s equation, Eq. (11.22a), is divided by ρ , we see that it becomes self-adjoint, and therefore, by the Sturm–Liouville theory, Section 10.2, the solutions are expected to be orthogonal — if we can arrange to have appropriate boundary conditions satisﬁed. To take care of the boundary conditions for a ﬁnite interval [0,a ], we introduce parameters a and ανm into the argument of J_{ν} to get J_{ν }(α_{νm} ρ/a ).Here a is the upper limit of the cylindrical radial coordinate ρ . From Eq. (11.22a),

Changing the parameter α_{νm} to α_{νn} , we ﬁnd that J_{ν} (α_{νn} ρ/a ) satisﬁes

Proceeding as in Section 10.2, we multiply Eq. (11.45) by J_{ν} (α_{νn }ρ/a ) and Eq. (11.45a) by J_{ν} (α_{νm} ρ/a ) and subtract, obtaining

Integrating from ρ = 0 to ρ = a , we obtain

Upon integrating by parts, we see that the left-hand side of Eq. (11.47) becomes F

This gives us orthogonality over the interval [0,a ].

**Normalization**

**Bessel Series**

If we assume that the set of Bessel functions J_{ν} (α_{νm} ρ /a ))(ν ﬁxed, m = 1, 2, 3,...)is complete, then any well-behaved but otherwise arbitrary function f(ρ ) may be expanded in a Bessel series (Bessel–Fourier or Fourier–Bessel)

**Example 11.2.1 ELECTROSTATIC POTENTIAL IN A HOLLOW CYLINDER **

From Table 9.3 of Section 9.3 (with α replaced by k ), our solution of Laplace’s equation in circular cylindrical coordinates is a linear combination of

The particular linear combination is determined by the boundary conditions to be satisﬁed.

Our cylinder here has a radius a and a height l . The top end section has a potential distribution ψ(ρ, ϕ ). Elsewhere on the surface the potential is zero. The problem is to ﬁnd the electrostatic potential

everywhere in the interior.

For convenience, the circular cylindrical coordinates are placed as shown in Fig. 11.3.

Since The z dependence becomes sinh kz, vanishing at z = 0. The requirement that ψ = 0 on the cylindrical sides is met by requiring the separation constant k to be

where the ﬁrst subscript, m, gives the index of the Bessel function, whereas the second subscript identiﬁes the particular zero of J_{m} .

The electrostatic potential becomes

Equation (11.56) is a double series: a Bessel series in ρ and a Fourier series in ϕ .

At z = l, ψ = ψ(ρ, ϕ ), a known function of ρ and ϕ . Therefore

The constants a_{mn} and b_{mn }are evaluated by using Eqs. (11.49) and (11.50) and the corresponding equations for sin ϕ and cos ϕ (Example 10.2.1 and Eqs. (14.2), (14.3), (14.15) to (14.17)). We ﬁnd 14

These are deﬁnite integrals, that is, numbers. Substituting back into Eq. (11.56), the series is speciﬁed and the potential ψ(ρ, ϕ , z) is determined.

**Continuum Form**

The Bessel series, Eq. (11.51) apply to expansions over the ﬁnite interval [0,a ].If a →∞, then the series forms may be expected to go over into integrals.

The discrete roots α_{νm }become a continuous variable α . A similar situation is encountered in the Fourier series,

For operations with a continuum of Bessel functions, J_{ν }(αρ ), a key relation is the Bessel function closure equation,

This may be proved by the use of Hankel transforms. An alternate approach, starting from a relation similar to Eq. (10.82), is given by Morse and Feshbach.

A second kind of orthogonality (varying the index) is developed for spherical Bessel functions.

**NEUMANN FUNCTIONS,BESSEL FUNCTIONS OF THE SECOND KIND**

From the theory of ODEs it is known that Bessel’s equation has two independent solutions.

Indeed, for nonintegral order ν we have already found two solutions and labeled them J_{ν} (x ) and J_{−ν} (x ), using the inﬁnite series (Eq. (11.5)). The trouble is that when ν is integral, Eq. (11.8) holds and we have but one independent solution. A second solution may be developed by the methods of Section 9.6. This yields a perfectly good second solution of Bessel’s equation but is not the standard form.

**Deﬁnition and Series Form **

As an alternate approach, we take the particular linear combination of J_{ν} (x ) and J_{−ν} (x )

This is the Neumann function (Fig. 11.5).^{15 }For nonintegral ν, N_{ν} (x ) clearly satisﬁes Bessel’s equation, for it is a linear combination of known solutions J_{ν} (x ) and J_{−ν} (x ).

**FIGURE 11.5 Neumann functions N _{0} (x ), N_{1} (x ), and N_{2} (x ).**

**Substituting the power-series Eq. (11.6) for n → ν (given in Exercise 11.1.7) yields**

for ν> 0. However, for integral ν, ν = n, Eq. (11.8) applies and Eq. (11.60) becomes indeterminate. The deﬁnition of N_{ν} (x ) was chosen deliberately for this indeterminate property. Again substituting the power series and evaluating N_{ν} (x ) for ν → 0 by l’Hôpital’s rule for indeterminate forms, we obtain the limiting value

for n = 0 and x → 0 , using

(11.63)

Equations (11.62) and (11.64) exhibit the logarithmic dependence that was to be expected.

This, of course, veriﬁes the independence of J_{n} and N_{n} .

**Other Forms**

As with all the other Bessel functions, N_{ν} (x ) has integral representations. For N_{0} (x ) we have

These forms can be derived as the imaginary part of the Hankel representations of Exercise 11.4.7. The latter form is a Fourier cosine transform.

To verify that N_{ν} (x ), our Neumann function (Fig. 11.5) or Bessel function of the second kind, actually does satisfy Bessel’s equation for integral n, we may proceed as follows.

L’Hôpital’s rule applied to Eq. (11.60) yields

Multiplying the equation for J_{−ν} by (−1)^{ν} , subtracting from the equation for J_{ν} (as suggested by Eq. (11.65)), and taking the limit ν → n, we obtain

(11.68)

It is seen from Eqs. (11.62) and (11.64) that N_{n} diverges, at least logarithmically. Any boundary condition that requires the solution to be ﬁnite at the origin (as in our vibrating circular membrane (Section 11.1)) automatically excludes N_{n} (x ). Conversely, in the absence of such a requirement, N_{n} (x ) must be considered.

To a certain extent the deﬁnition of the Neumann function N_{n} (x ) is arbitrary. Equations (11.62) and (11.64) contain terms of the form an J_{n} (x ). Clearly, any ﬁnite value of the constant an would still give us a second solution of Bessel’s equation. Why should an have the particular value implicit in Eqs. (11.62) and (11.64)? The answer involves the asymptotic dependence developed in Section 11.6. If J_{n} corresponds to a cosine wave, then N_{n} corresponds to a sine wave. This simple and convenient asymptotic phase relationship is a consequence of the particular admixture of J_{n }in N_{n} .

**Recurrence Relations**

Substituting Eq. (11.60) for N_{ν} (x ) (nonintegral ν ) into the recurrence relations (Eqs. (11.10) and (11.12) for J_{n} (x ), we see immediately that N_{ν} (x ) satisﬁes these same recurrence relations. This actually constitutes another proof that N_{ν} is a solution. Note that the converse is not necessarily true. All solutions need not satisfy the same recurrence relations.

**Wronskian Formulas **

From Section 9.6 and Exercise 10.1.4 we have the Wronskian formula17 for solutions of the Bessel equation,

(11.69)

in which A_{ν} is a parameter that depends on the particular Bessel functions u_{ν} (x ) and v_{ν} (x ) being considered. A_{ν} is a constant in the sense that it is independent of x . Consider the special case

Since A_{ν} is a constant, it may be identiﬁed at any convenient point, such as x = 0. Using the ﬁrst terms in the series expansions (Eqs. (11.5) and (11.6)), we obtain

Substitution into Eq. (11.69) yields

using Eq. (8.32). Note that A_{ν} vanishes for integral ν , as it must, since the nonvanishing of the Wronskian is a test of the independence of the two solutions. By Eq. (11.73), J_{n }and J_{−n} are clearly linearly dependent.

Using our recurrence relations, we may readily develop a large number of alternate forms, among which are

Many more will be found in the references given at chapter’s end.

Wronskians were of great value in two respects: (1) in establishing the linear independence or linear dependence of solutions of differential equations and (2) in developing an integral form of a second solution. Here the speciﬁc forms of the Wronskians and Wronskian-derived combinations of Bessel functions are useful primarily to illustrate the general behavior of the various Bessel functions. Wronskians are of great use in checking tables of Bessel functions.

**Example 11.3.1 COAXIAL WAVE GUIDES**

We are interested in an electromagnetic wave conﬁned between the concentric, conducting cylindrical surfaces ρ = a and ρ = b. To go from the standing wave of these examples to the traveling wave here, we let A = iB, A = a_{mn} ,B = b_{mn} in Eq. (11.40a) and obtain

For the coaxial wave guide one generalization is needed. The origin, ρ = 0, is now excluded (0 <a ≤ ρ ≤ b). Hence the Neumann function N_{m} (γ ρ ) may not be excluded. E_{z} (ρ ,ϕ ,z,t ) becomes

With the condition

H_{z} = 0, (11.80)

we have the basic equations for a TM (transverse magnetic) wave.

The (tangential) electric ﬁeld must vanish at the conducting surfaces (Dirichlet boundary condition), or

These transcendental equations may be solved for γ(γ_{mn} ) and the ratio c_{mn} /b_{mn} .From Example 11.1.2,

Since k^{2} must be positive for a real wave, the minimum frequency that will be propagated (in this TM mode) is

ω = γc, (11.84)

with γ ﬁxed by the boundary conditions, Eqs. (11.81) and (11.82). This is the cutoff frequency of the wave guide.

There is also a TE (transverse electric) mode, with E_{z} = 0 and Hz given by Eq. (11.79).

Then we have Neumann boundary conditions in place of Eqs. (11.81) and (11.82). Finally, for the coaxial guide (not for the plain cylindrical guide, a = 0), a TEM (transverse electromagnetic) mode, E_{z} = H_{z} = 0, is possible. This corresponds to a plane wave, as in free space.

To conclude this discussion of Neumann functions, we introduce the Neumann function N_{ν} (x ) for the following reasons:

1. It is a second, independent solution of Bessel’s equation, which completes the general solution.

2. It is required for speciﬁc physical problems such as electromagnetic waves in coaxial cables and quantum mechanical scattering theory.

3. It leads to a Green’s function for the Bessel equation.

4. It leads directly to the two Hankel functions .

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