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**BESSEL FUNCTIONS OF THE FIRST KIND, J _{ν} (x ) **

Bessel functions appear in a wide variety of physical problems. Separation of the Helmholtz, or wave, equation in circular cylindrical coordinates led to Bessel’s equation. we will see that the Helmholtz equation in spherical polar coordinates also leads to a form of Bessel’s equation. Bessel functions may also appear in integral form — integral representations. This may result from integral transforms or from the mathematical elegance of starting the study of Bessel functions with Hankel functions.

Bessel functions and closely related functions form a rich area of mathematical analysis with many representations, many interesting and useful properties, and many interrelations.

**Generating Function for Integral Order **

Although Bessel functions are of interest primarily as solutions of differential equations, it is instructive and convenient to develop them from a completely different approach, that of the generating function.1 This approach also has the advantage of focusing on the functions themselves rather than on the differential equations they satisfy. Let us introduce a function of two variables,

(11.1)

Expanding this function in a Laurent series, we obtain

(11.2)

It is instructive to compare Eq. (11.2) with the equivalent Eqs. (11.23) and (11.25).

The coefﬁcient of t^{n} ,J_{n} (x ), is deﬁned to be a Bessel function of the ﬁrst kind, of integral order n. Expanding the exponentials, we have a product of Maclaurin series in xt /2 and −x/2t , respectively,

Here, the summation index r is changed to n, with n = r − s and summation limits n =−s to ∞, and the order of the summations is interchanged, which is justiﬁed by absolute convergence. The range of the summation over n becomes −∞ to ∞, while the summation over s extends from max(−n, 0) to ∞. For a given s we get t^{n} (n ≥ 0) from r = n + s :

The coefﬁcient of t^{n} is then ^{2}

This series form exhibits is behavior of the Bessel function J_{n} (x ) for small x and permits numerical evaluation of J_{n} (x ). The results for J_{0} ,J_{1} , and J_{2} are shown in Fig. 11.1. the error in using only a ﬁnite number of terms of this alternating series in numerical evaluation is less than the ﬁrst term omitted. For instance, if we want J_{n} (x )

to ±1% accuracy, the ﬁrst term alone of Eq. (11.5) will sufﬁce, provided the ratio of the second term to the ﬁrst is less than 1% (in magnitude) or x< 0.2(n + 1)^{1/2} . The Bessel functions oscillate but are not periodic — except in the limit as x →∞ (Section 11.6). The amplitude of J_{n} (x ) is not constant but decreases asymptotically as x^{ −1/2} . (See Eq.(11.137) for this envelope.)

For n< 0, Eq. (11.5) gives

Since n is an integer (here), (s − n)!→∞ for s = 0,...,(n − 1). Hence the series may be considered to start with s = n. Replacing s by s + n, we obtain

showing immediately that J_{n} (x ) and J−n (x ) are not independent but are related by

These series expressions (Eqs. (11.5) and (11.6)) may be used with n replaced by ν to deﬁne J_{ν }(x ) and J_{−ν} (x ) for nonintegral ν (compare Exercise 11.1.7).

**Recurrence Relations**

The recurrence relations for J_{n} (x ) and its derivatives may all be obtained by operating on the series, Eq. (11.5), although this requires a bit of clairvoyance (or a lot of trial and error). Veriﬁcation of the known recurrence relations is straightforward, Exercise 11.1.7.

Here it is convenient to obtain them from the generating function, g(x , t ). Differentiating both sides of Eq. (11.1) with respect to t , we ﬁnd that

and substituting Eq. (11.2) for the exponential and equating the coefﬁcients of like powers of t ,^{3} we obtain

This is a three-term recurrence relation. Given J_{0} and J_{1} , for example, J_{2} (and any other integral order J_{n} ) may be computed.

Differentiating Eq. (11.1) with respect to x ,wehave

Again, substituting in Eq. (11.2) and equating the coefﬁcients of like powers of t , we obtain the result

(11.12)

As a special case of this general recurrence relation,

(11.13)

Adding Eqs. (11.10) and (11.12) and dividing by 2, we have

(11.14)

Multiplying by x^{ n} and rearranging terms produces

(11.15)

(11.16)

Multiplying by x −n and rearranging terms, we obtain

(11.17)

**Bessel’s Differential Equation **

Suppose we consider a set of functions Z_{ν} (x ) that satisﬁes the basic recurrence relations (Eqs. (11.10) and (11.12)), but with ν not necessarily an integer and Z_{ν} not necessarily given by the series (Eq. (11.5)). Equation (11.14) may be rewritten (n → ν) as

(11.18)

On differentiating with respect to x ,wehave

(11.19)

Multiplying by x and then subtracting Eq. (11.18) multiplied by ν gives us

(11.20)

Now we rewrite Eq. (11.16) and replace n by ν − 1:

(11.21)

Using Eq. (11.21) to eliminate Z_{ν −1} and Z_{ν}′_{ −1 }from Eq. (11.20), we ﬁnally get

(11.22)

which is Bessel’s ODE. Hence any functions Z_{ν} (x ) that satisfy the recurrence relations (Eqs. (11.10) and (11.12), (11.14) and (11.16), or (11.15) and (11.17)) satisfy Bessel’s equation; that is, the unknown Z_{ν} are Bessel functions. In particular, we have shown that the functions J_{n} (x ), deﬁned by our generating function, satisfy Bessel’s ODE. If the argument is kρ rather than x , Eq. (11.22) becomes

**Integral Representation**

A particularly useful and powerful way of treating Bessel functions employs integral representations. If we return to the generating function (Eq. (11.2)), and substitute t = e^{iθ} ,we get

in which we have used the relations

and so on.

In summation notation,

equating real and imaginary parts of Eq. (11.23).

By employing the orthogonality properties of cosine and sine,

in which n and m are positive integers (zero is excluded), we obtain

If these two equations are added together,

As a special case (integrate Eq. (11.25) over (0,π ) to get)

Noting that cos(x sin θ) repeats itself in all four quadrants, we may write Eq. (11.30) as

On the other hand, sin(x sin θ) reverses its sign in the third and fourth quadrants, so

Adding Eq. (11.30a) and i times Eq. (11.30b), we obtain the complex exponential representation

This integral representation (Eq. (11.29)) may be obtained somewhat more directly by employing contour integration (compare Exercise 11.1.16).^{6} Many other integral representations exist (compare Exercise 11.1.18).

**Example 11.1.1 FRAUNHOFER DIFFRACTION,CIRCULAR APERTURE **

In the theory of diffraction through a circular aperture we encounter the integral

for φ, the amplitude of the diffracted wave. Here θ is an azimuth angle in the plane of the circular aperture of radius a , and α is the angle deﬁned by a point on a screen below the circular aperture relative to the normal through the center point. The parameter b is given by

(11.32)

with λ the wavelength of the incident wave. The other symbols are deﬁned by Fig. 11.2.

From Eq. (11.30c) we get^{8}

(11.33)

Equation (11.15) enables us to integrate Eq. (11.33) immediately to obtain

(11.34)

Note here that J_{1} (0) = 0. The intensity of the light in the diffraction pattern is proportional to φ^{2 }and

(11.35)

From Table 11.1, which lists the zeros of the Bessel functions and their ﬁrst derivatives, Eq. (11.35) will have a zero at

(11.36)

or

(11.37)

For green light, λ = 5.5 × 10^{−5} cm. Hence, if a = 0.5cm,

α ≈ sin α = 6.7 × 10^{−5} (radian) ≈ 14 seconds of arc, (11.38)

which shows that the bending or spreading of the light ray is extremely small. If this analysis had been known in the seventeenth century, the arguments against the wave theory of light would have collapsed. In mid-twentieth century this same diffraction pattern appears in the scattering of nuclear particles by atomic nuclei — a striking demonstration of the wave properties of the nuclear particles.

A further example of the use of Bessel functions and their roots is provided by the electromagnetic resonant cavity (Example 11.1.2) and the example and exercises of Section 11.2.

**Example 11.1.2 CYLINDRICAL RESONANT CAVITY**

The propagation of electromagnetic waves in hollow metallic cylinders is important in many practical devices. If the cylinder has end surfaces, it is called a cavity. Resonant cavities play a crucial role in many particle accelerators.

**11.1 Bessel Functions of the First Kind, J _{ν }(x )**

**FIGURE 11.3 Cylindrical resonant cavity.**

We take the z-axis along the center of the cavity with end surfaces at z = 0 and z = l and use cylindrical coordinates suggested by the geometry. Its walls are perfect conductors, so the tangential electric ﬁeld vanishes on them (as in Fig. 11.3):

E_{z} = 0 = E_{ϕ} for ρ = a, E_{ρ }= 0 = E_{ϕ} for z = 0,l.

Inside the cavity we have a vacuum, so ε_{0} µ_{0} = 1/c^{2} . In the interior of a resonant cavity, electromagnetic waves oscillate with harmonic time dependence e^{−iωt }, which follows from separating the time from the spatial variables in Maxwell’s equations (Section 1.9), so

With ∇ · E = 0 (vacuum, no charges) and Eq. (1.85), we obtain for the space part of the electric ﬁeld

which is called the vector Helmholtz PDE.The z-component ( E_{z} , space part only) satisﬁes the scalar Helmholtz equation,

(11.39)

The transverse electric ﬁeld components E_{⊥ }= (E_{ρ} ,E_{ϕ} ) obey the same PDE but different boundary conditions, given earlier. Once E_{z }is known, Maxwell’s equations determine Eϕ fully.

This implies

Here, k^{2 }is a separation constant, because the left- and right-hand sides depend on different variables. For w(z) we ﬁnd the harmonic oscillator ODE with standing wave solution (not transients) that we seek,

w(z) = A sin kz + B cos kz,

with A, B constants. For v(ρ , ϕ ) we obtain

where the separation constant m^{2} must be an integer, because the angular solution φ = e^{imϕ }of the ODE

must be periodic in the azimuthal angle.

This leaves us with the radial ODE

Dimensional arguments suggest rescaling ρ → r = γρ and dividing by γ^{2} , which yields

This is Bessel’s ODE for ν = m. We use the regular solution J_{m} (γ ρ ) because the (irregular) second independent solution is singular at the origin, which is unacceptable here. The complete solution is

where the constant γ is determined from the boundary condition E_{z} = 0 on the cavity surface ρ = a, that is, that γa be a root of the Bessel function J_{m} (see Table 11.1). This gives a discrete set of values γ = γ_{mn} , where n designates the nth root of J_{m }(see Table 11.1).

For the transverse magnetic or TM mode of oscillation with H_{z} = 0 Maxwell’s equations imply. (See again Resonant Cavities in J. D. Jackson’s Electrodynamics in Additional Readings.)

with

(11.43)

where α_{mn }is the nth zero of J_{m} . The general solution

with constants B_{mnp }, now follows from the superposition principle.

The result of the two boundary conditions and the separation constant m^{2} is that the angular frequency of our oscillation depends on three discrete parameters:

These are the allowable resonant frequencies for our TM mode.

**Alternate Approaches**

Bessel functions are introduced here by means of a generating function, Eq. (11.2). Other approaches are possible. Listing the various possibilities, we have:

- Generating function (magic), Eq. (11.2).
- Series solution of Bessel’s differential equation,
- Contour integrals: Some writers prefer to start with contour integral deﬁnitions of the Hankel functions, and develop the Bessel function J
_{ν}(x) from the Hankel functions. - Direct solution of physical problems: Example 11.1.1. Fraunhofer diffraction with a circular aperture, illustrates this. Incidentally, Eq. (11.31) can be treated by series expansion, if desired. Feynman10 develops Bessel functions from a consideration of cavity resonators.

In case the generating function seems too arbitrary, it can be derived from a contour integral, or from the Bessel function recurrence relations,

Note that the contour integral is not limited to integer ν , thus providing a starting point for developing Bessel functions.

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